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NATIONAL BUREAU OF STANDARDS REPORT 8552 Progress Report on Viscoelastic Behavior of Dental Amalgam U.S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS

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Page 1: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

NATIONAL BUREAU OF STANDARDS REPORT

8552

Progress Report

on

Viscoelastic Behavior

of Dental Amalgam

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Page 2: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

maximum application of the physical and engineering sciences to the advancement of technology in

industry and commerce. Its responsibilities include development and maintenance of the national stand-

ards of measurement, and the provisions of means for making measurements consistent with those

standards; determination of physical constants and properties of materials; development of methodsfor testing materials, mechanisms, and structures, and making such tests as may be necessary, particu-

larly for government agencies; cooperation in the establishment of standard practices for incorpora-

tion in codes and specifications; advisory service to government agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; assistance

to industry, business, and consumers in the development and acceptance of commercial standards andsimplified trade practice recommendations; administration of programs in cooperation with United

States business groups and standards organizations for the development of international standards of

practice; and maintenance of a clearinghouse for the collection and dissemination of scientific, tech-

nical, and engineering information. The scope of the Bureau’s activities is suggested in the following

listing of its four Institutes and their organizational units.

Institute for Basic Standards. Electricity. Metrology, Heat. Radiation Physics. Mechanics. Ap-

plied Mathematics. Atomic Physics. Physical Chemistry, Laboratory Astrophysics.* Radio Stand-

ards Laboratory: Radio Standards Physics; Radio Standards Engineering,** Office of Standard Ref-

erence Data.

Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Mate-

rials. Reactor Radiations. Cryogenics.** Office of Standard Reference Materials.

Central Radio Propagation Laboratory.** Ionosphere Research and Propagation. Troposphere

and Space Telecommunications. Radio Systems, Upper Atmosphere and Space Physics.

Institute for Applied Technology. Textiles and Apparel Technology Center. Building Research.

Industrial Equipment. Information Technology. Performance Test Development. Instrumentation.

Transport Systems. Office of Technical Services. Office of Weights and Measures. Office of Engineer-

ing Standards. Office of Industrial Services.

* NBS Group, Joint Institute for Laboratory Astrophysics at the University of Colorado.** Located at Boulder, Colorado.

Page 3: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

311 . 05 -20- 31 1 0560 J une 30, 1 964

NBS REPORT

8552

Progress Report

on

Viscoelastic Behavior

of Dental Amalgam

By

P. L. Oqlesby*G. DicKSon*

R. D. Davenport*M. S. Rodriguez**W. T. Sweeney***

* Dental Research Section, National Bureau of Standards, Washington, D. C. 20234.** Guest Worker, National Bureau of Standards. Presently teaching dental materials at

Loyola University Dental School, New Orleans, Louisana.*** Chief, Dental Research Section, National Bureau of Standards, Washington, D. C. 20234.

This investigation was conducted at the National Bureau of Standards in cooperationwith the Council on Dental Research of the American Dental Association, the ArmyDental Corps, the Dental Sciences Division of the School of Aerospace Medicine, USAF,and the Veterans Administration.

IMPORTANT NOTICE

Approved for public release by thedirector of the National Institute of

Standards and Technology (NIST)

on October 9, 2015

<NB^U.S. DEPARTMENT OF COMMERCE

ccounting documents intended

ected to additional evaluation

ing of this Report, either in

fice of the Director, National

e Government agency for which

s tor its own use.

NATIONAL BUREAU OF STANDfor use within the Government. Befc

and review. For this reason, the pul

whole or in part, is not authorized

Bureau of Standards, Washington, D

the Report has been specifically prei

NATIONAL BUREAU OF STANDARDS

Page 4: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

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Page 5: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

Viscoelastic Behaviorof Dental Amalgam

Abstract

Measurements made on dental amalgam in tension indicate that amalgam exhibitsthree types of viscoelastic phenomena: (l) instantaneous elastic strain,(2) retarded elastic strain (transient creep) and (3) viscous strain (steadystate creep). The combination of elastic plus retarded strain can be repre-sented by an equation of the form € = Act + B^cj^where A and B are functions oftime but not of the stress,

^

ct. The viscous strain rate can be representedby an equation of the form Gv = Kct™ where K and m are constants of the ma-terial. By applying a nonlinear generalization of the Boltzmann superposi-tion principle to a general equation describing the creep behavior of amal-gam, the results of creep tests can be directly related to the results ofstress-strain tests.

1, Introduction

There is no information in the literature on the viscoelastic behavior of dentalamalgam except the limited amount that could be obtained from stress-strain relation-ships [1,2]. The stress-strain data indicate that amalgam exhibits a nonlinear re-lation between stress and strain over the entire range of stress investigated; thishas been found to be true in both tension and compression. It has been further notedthat the shape of the stress-strain curves vary with the strain rate indicating thatthe strain developed is not only dependent on the applied stress but also upon timeof application of the stress. The results of tensile stress-strain investigation byRodriguez [1] indicated that dental amalgam might be a viscoelastic material. Itwas, therefore, the objective of this study (l) to make an exploratory investigationof the types of viscoelastic phenomena exhibited by dental amalgam in tension (2) todescribe the viscoelastic phenomena exhibited by amalgam in terms of available visco-elastic theory and (.3 ) to make available a practical example and method of applicationof viscoelastic theory to a dental material.

2. Theory

In general a strain-hardened material will exhibit at least one or more of threetypes of viscoelastic phenomena; (l) elastic behavior (2) viscous behavior (steady-state creep) and (3) retarded elastic behavior (transient creep).

The elastic behavior is a linear function of the applied stress and completelyrecoverable; that is to say the elastic strain developed in a material is directlyproportional to the applied stress and disappears instantaneously upon removal of thestress. The instantaneous elastic response of a material is commonly ' represented bya spring as shown in Figure 1-A.

€e= Jocr (1)

where

€g = the elastic strain developed in the material

CT = the applied stress

Jq = the proportionality constant called the elastic compliance which is the re-ciprocal of the elastic modulus.

In the case of viscous behavior (steady-state creep), the strain developed in thematerial is a function of both the applied stress and the time of application of thestress, but the viscous strain is nonrecoverable upon removal of the stress, that is tosay it is permanent. The viscous strain is a linear function of time, and the viscousstrain rate for any given stress value is a constant; on the other hand the viscousstrain may or may not be a linear function of the applied stress. The viscous responseof a material can be represented by the behavior of a dashpot as shown in Figure 1-B.

The viscous strain rate for a material exhibiting a linear relation to theapplied stress is as follows: = 1 (^-cto)

1

Page 6: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

where

€y is the viscous strain rate

T] Is the coefficient of viscosity of the material

CT Is the applied stress

a Is the applied stress level at which the material first begins to exhibitviscous flow and Is called the yield stress.

A material exhibiting viscous flow In accordance with the above relation Is said toexhibit Ideal plastic behavior (Bingham behavior) while a material which obeys theabove equation in its viscous flow but has a value of Cq which is zero. Is said to bea material which exhibits Newtonian flow. Materials demonstrating viscous flow whichIs a nonlinear function of the applied stress; usually obey one of the following twoequations in their viscous response

6^ = K (ct - aQ)*^ or = A slnh Be (3)^ (^)

where K and m, and A and B are constants at any given temperatures, A materialexhibiting nonlinear viscous response described by the first equation Is said to bepseudo plastic [3] In its viscous behavior in either case whether CTq Is or Is not zero.On the other hand a material demonstrating nonlinear viscous response according to thesecond equation does not have a yield stress ctq and Is said to exhibit viscous flow(plastic flow) In accordance with the Kauzmann rate theory [4], This type of viscousflow occurs In metals at temperatures near their melting points [5 j>6,7,8],

The retarded elastic response of a material may be represented by a single modelor by either a finite or an Infinite series of models called Voigt elements. A Voigtelement Is shown In Figure 1-C.

Retarded elastic behavior (transient creep) which can be described by means of asingle Voigt element Is represented by the following equation:

J Cl-e"^/r) a (5)

where

^R

CT

J

T

is

Is

is

Is

the retarded elastic

the applied stress

the retarded elastic

the retardation time

strain developed

compliance

of the material.

in the material

Retarded elastic behavior corresponding to a finite series of Voigt elements is repre-sented by the following equation:

n

^ ^ JlCl-e"^/'^0 (6)

where

is the retarded elastic strain developed In the material

CT Is the applied stress

Is the retarded elastic compliance of each Voigt element

Is the retardation time of each Voigt element

Retarded elastic response represented by an infinite number of Voigt elements Isdescribed by the following equation:

^R J(t) T(7)

o

2

Page 7: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

In all of the above equations for the representation of retarded elastic strain,the relation between the retarded strain and the applied stress Is assumed to be llneaijhowever. In many materials this relation has been found to be nonlinear. The mathema-tical description of the nonlinear case has been treated In various publications [9, 10],For example. If the retarded elastic strain (transient creep) Is related to the appliedstress, by a second degree equation In which the time response can be represented by anInfinite number of Voigt elements, such a retarded elastic response can be describedby the following equation:

( 1-e"^/ d T + B(y) (l-e"^'^Y) d ( 8 )

where

Is the retarded elastic strain

CT Is the applied stress

J(t) Is the continuous distribution of retarded elastic compliance as a functionof the variable retardation time t of the continuous distribution of Voigtresponses In the linear stress response element of the material

B(y) Is the continuous distribution of retarded elastic compliances as a functionof the variable retardation time yofthe continuous, distribution of Voigtelements In the nonlinear second degree stress response of the material.

In a material which exhibits all three types of behavior (elastic, viscous, andretarded elastic), and In which the three types of behavior are additive (obey asuperposition relation), the creep strain developed In the material at any time t underan applied stress a may be related to the strains due to the Individual behaviors bythe following equations:

^ ^o + ^v ^5)

where

€ Is the creep strain developed In the material at any time t under an appliedconstant stress

^0 Is the Instantaneous elastic strain developed In the material

€r Is the retarded elastic strain developed In the material

Is the viscous strain developed In' the material.

These Individual responses as was noted earlier may be linearly or nonllnearly relatedto the stress. For example. In a material which exhibits all three phenomena linearlyrelated to the applied stress then using the linear equation of each response givenearlier, the above equation takes the form:

CO

% '-^o r J(t) (l-e'^^T ) d T + — (10)

J T1

o

Next, consider an example of a material which exhibits all three types of behavior, bit

In which the viscous strain developed Is nonllnearly related to the applied stress asfollows

:

€v= K (cr - Cq) "^t

( 11 )

while the retarded elastic strain Is nonllnearly related to the applied stress by anequation of second degree as given earlier and the Instantaneous Is assumed to be

linear; thus, the above superposition equation for creep strain takes the followingform:

€ Jo^|j(t) (l-e-Vr)

d T + a2

o

’b(y) (1-6-^^y) ^^

4- K(ct (12 )

3

o

Page 8: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

Where the creep strain is linearly related to the applied stress, the strain canbe written as a product of a general creep compliance, which Is a function of timeonly, and the applied stress as given by the following equation:

€ = J(t)a (13)

where

G Is the creep strain developed

CT Is the applied stress

J(t) Is the general creep compliance

For example, in the linear creep behavior as seen in equation (10) , the general creepcompliance J(t) is of the form:

j(t; J(' L-e (14)

In the case that creep strain Is linearly related to the applied stress, thestress-strain behavior for a material may be related to the creep behavior by meansof the linear superposition principle developed by Boltzmann [ 11 ] as shown below

:

^tE (crt)

=JJ(T- 9 ) d a (6) (I5)

o

orT

E (T)- Jj(T-e) d 9 (16)

o

where

a is the applied stress related to the experimental time 9 in a prescribedexperimental functional relation.

e(t) is the strain of a stress-strain relation measured at a time T subsequentto the variable time 9.

- d 9 Is the Increment of stress from time 9 to time ( 9 + d 9 )

.

For example, applying the Boltzmann superposition principle to the example ofcreep given In equation (10) gives the following relation between creep behavior andthe strain measured In a stress-strain test [12]

:

E (T)

T-9

(l-e- ) d T d 9 +d 9

•(e) d e (17)

0 o o

An excellent discussion of the application of the Boltzmann superposition principleas applied to linear viscoelastic phenomena is given by Leaderman [13]. However, ifa material exhibits a nonlinear relation in Its retarded strain ( transient creep) andthe applied stress, as discussed earlier, the linear Boltzmann superposition principlecannot be applied In Its present form to relate strain of a stress-strain test on thematerial to the creep strain. A generalization of Boltzmann ' s superposition principledeveloped by Nakada [10] may be used to relate nonlinear retarded elastic creep be-havior to the stress-strain behavior of the material as given:

(o-)

j

T)

t(T-9) d o- (9

(T) cr(T) CT (T)

.(T) a(T)

•<- r f (T-9) $ (T-0) d CT (e) d CT (0)

E(T-9 )E(T-0) E(T-a) d ct (9) d a (0) d ct (a) (18)

o o

4

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Now for example, consider the application of the nonlinear superposition principle ofNakada as applied, to the case of retarded strain (transient creep) related to theapplied stress by a second degree equation as shown earlier:

'2

now

t(t:

00

J(t) (l-e~^^T^ d T + cT^

rB(v) (l-e‘*/v) d V

0

)

J(t), (l-e“^^T^ d

(19)

and00

?(t) = rB(Y) (l-e“^/v) d

where t (t) and $(t) are the respective first degree and second degree retardationfunctions as given In the nonlinear superposition principle. Therefore, applying thenonlinear superposition principle to the above retarded creep example one has:

/ ^ p=Kt)» _T-e .

{a{T)J=J

['j(T)(l-e^ ) d t d ct (9

(T)^(T) » ®^

T-0

(y) (l-e~^

^dY'^Vdcr (0) d a ( 0 ) (20)

0 0

Considering an example of a material which exhibits all three phenomena but In which

the creep strain Is a nonlinear function of the applied stress according to equation(12), then applying the nonlinear superposition principle, the stress-strain behaviorof the material may be related to the creep behavior by the following equation:

E •= CT Jq + r r J( t) (l-e'^^^) d T d a (t)eo

,T T[T-0] . _[T-0 k

B(y) (l-e Y J (l-e V J d y d cr (0) d a (0

+jK a^{t) a t

o

(21)

3. Materials

The specimens were prepared using a commercial alloy for dental amalgam certifiedto comply with American Dental Association Si5'ecificatlon No. 1, This alloy (composi-tion approximately Ag 7O/0 , Sn 26^, Cu -Zn 0.5^) was mixed with mercury and con-densed Into a mold as described by Rodriguez and Dickson [1] to produce a specimenwith dimensions as shown in Figure 2. Speclmacs were aged for at least one week to ob-tain essentially full mechanical strength [14].

4. Procedure

The dumbbell-shaped specimen was placed in the grips and Tuckerman optical straingages were mounted on opposite sides of the specimen as shown In Figure 3. To obtainthe creep curve, readings were taken on the strain gages, a weight was suspended fromthe lower grip and a second strain gage reading was taken Immediately. Strain read-ings were then made at 15 second Intervals for 4 minutes and at Increasingly longerintervals until the strain rate became constant (usually after approximately 1.5 hours).At the end of this period, the load was removed, a strain reading was taken Immediatelyand the recovery curve was followed by reading first at 15 second Intervals and then at

5

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longer Intervals until the strain became constant.

Strain readings obtained on the two sides of the specimen were averaged and strainwas plotted against time to obtain the loaded creep and unloaded recovery curves.Readings on the strain gages were normally made to the nearest 2 X 10“5 inch. Since agage length of 0.25 inch was used this is equivalent to a strain of 8 X 10-5. Thusin the results given below differences in strain of 1 X 10“^ are approximately equalto the minimum reading difference.

2Loads placed on the specimen (with a nominal 0.01 in cross sectional area) varied

from 5 to 40 lbs, giving stresses from approximately 500 to 4^000 psi. Most specimenswere used for several runs, first at high and then at lower stresses. The firstloaded creep run was , considered a strain hardening treatment and data obtained onthese runs were not used in the calculation of results other than for viscous strain rate.All runs were made at 23 ± 1°C.

5. Results and Discussion

The creep curves (both loaded and recovery curves) of strain hardened dentalamalgam as shown for a number of different stress levels in Figure 4 indicate that atroom temperature amalgam exhibits three different types of viscoelastic phenomena:(l) Instantaneous elastic strain (2) retarded elastic strain (transient creep) and( 3 ) viscous strain (steady-state creep).

The viscous strain rate was determined from the loaded portion of the creep curveby taking the slope of the straight line portion of the curve, and was also determinedfrom the recovery portion of the creep curve by dividing the value of the recoverystrain (the permanent strain in the specimen) by the total time the load was on thespecimen. The viscous strain rates for any given creep curve as calculated from theloaded and recovery portions of the curve were found to agree fairly well as shownin Table 1. The log of the viscous strain rate was found to be a linear function ofthe log of the applied stress, as shown in Figure 5j thus the viscous strain ratecould be related to the applied stress by the following equation:

€^=- K cr™( 22 )

where

is the viscous strain rate

CT is the applied stress

K and m are constant of the material

The value of m for amalgam is the value of the slope of the curve in Figure 5.While the value of K is the antilog of the viscous strain rate value at a value ofapplied stress a of 1 psi. The values of K and m for the dental amalgam used in thisinvestigation were found to be K equals 2.85 and 4.98 X 10-19 and m equals 3.99 and3.92 from loaded and unloaded data respectively.

The strain developed in amalgam due to the other two phenomena (l) Instantaneouselastic strain and (2) retarded strain can be determined from the strain recoverysince these two types of strain are recoverable while the viscous strain is not. Thus,at any given load the strain values taken from the creep curve after the sample hasbeen unloaded (that is in the recovery portion of the creep curve) are subtractedfrom the strain value on the creep curve at the instant Just before unloading of thespecimen. This difference is plotted against recovery time tj_ = T^ - T^ where T^ is

the time at which the specimen was unloaded and Tj_ is the time of the strain value onthe recovery portion of the curve. These difference values, 5', are seen plottedagainst the recovery time for each load or stress in Figure 6. These plots are ameasure of the combination of the elastic and retarded elastic strain behavior of den-tal amalgam as a function of time for various stress levels. In theory the same plotmay be obtained from the loaded portion of the creep curve bj"- taking values off theloaded creep curve and subtracting the viscous strain accumulated in the specimen atthat time. The accumulated viscous strain at any time may be calculated by multiplyingthe viscous strain rate by the time corresponding to that value on the creep curve.Thus the difference between the creep curve value on the loaded protion and the viscousstrain value at a corresponding time is a measure of the combination of the instantan-eous and retarded elastic strain. However, a small error in the viscous strain ratecauses a large error in the difference value. Therefore, the plot of the combination

6

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of elastic and retarded elastic strain versus time as obtained from the loaded creepcurve is subject to large possible error; for this reason the combination elastic be-havior is obtained from the recovery portion of the creep curves.

The combination elastic strain (instantaneous and retarded elastic strain) valuesare seen in Table 2 tabulated against time for various applied stress levels. Thecombination strain becomes asymptotic with time in accordance with theory as seen inFigure 6. The combination elastic strain values were plotted as a function of thevarious stress levels for corresponding time; as shown in Figure 7. The combinationelastic strain is seen to be a nonlinear function of the applied stress. When the com-bination strain values were divided by their corresponding stresses and then plottedagainst the corresponding stress for a fixed time a linear plot was obtained for eachfixed time as illustrated in Figure 8; this result indicated that the combination elas-tic behavior of amalgam as a function of applied stress under the test conditions couldbe represented by an equation of the form:

€'= A(t) a + B2(t) a2 (23)

where

e' is the combination of elastic and retarded elastic strain

a is the applied stress

A(t) and B(t) are constants for any given time value and are functions of time butnot of stress. The value of A(t) for any time value is the intercept at a = 0 of theplot for that time value as shown in Figure 8 while B2(t) is the slope of the straightline for that time value. It is also hoted in Figure 8 that as a function of stressthe combination strain divided by the stress is a straight line for all values of t.This indicated that over all ranges of t the combination elastic strain obeys the samefunctional relation to the stress. Thus from Figure 6 it was concluded that the re-tarded elastic strain as a function of time could be described by means of conventionalviscoelastic theory assuming either a finite or infinite number of Voigt elements whilethe combination elastic strain being a nonlinear function of stress indicated the needfor the use of nonlinear viscoelastic theory as developed first by Leaderman[9l andlater more generally by Nakada [10]. Therefore, the values for A(t) and B2(t) weredetermined by fitting curves to the data by the method of least squares and were tabu-lated as a function of time as shown in Table 3. The A(t) values were plotted as afunction of the corresponding t values as shown in Figure 9. The A(t) values are seento approach an asymptote as t— thus when the values of the asymptote minus theA(t) values are plotted as a function of the corresponding time values. Figure 10,the curve is seen to fall off in a complicated exponential behavior which could be re-presented by viscoelastic theory assuming an infinite number of Voigt elements or acontinuous spectrum. Thus, it has been, assumed that A(t) could be represented by thefollowing equation from linear viscoelastic theory since A(t) is the linear term instress

:

00

A(t) = Jq tJj(t)0

A ( t )= Jq +

CO

r j(t)j0

00

A(t) = Agg -\

J(t)

1

>“

Aas- Mt) ]J(t) e

o

(24)

( 25 )

(26)

( 27 )

where

A(t) is the creep compliance term which is linear in stress

Jq is Instantaneous elastic compliance

J(t) Is the retarded elastic compliance spectrum as a function of theretardation time t

7

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Agg is the asymptote value of A(t) as t becomes very large.

Thus, equation (26) indeed does describe the asymptotic behavior of A(t) as afunction of time as seen in Figure 9 while equation (27) could be used to describethe complicated exponential-like behavior seen in Figure 10. The linear creep compli-ance term A(t) is plotted as a function of log t to obtain a sigmoidal curve as shownin Figure 11. The first plateau of the sigmoidal plot at very small values of tshould correspond to Jq (the instantaneous elastic compliance )[ 15 ] , however, thecurve does not extend to short enough time values to determine the value of the plateauwhich would be Jq

.

Therefore, it is concluded that Instantaneous elastic behavior ofamalgam cannot be separated from the combination elastic behavior and thus is indeter-minate from the data obtained in this investigation. The plot in Figure 11 indicatesthe need for improved experimentation on anjalgam in which strain measurements can bemade at very short times after loading or removal of the load from the specimen.

Next the nonlinear creep compliance term B'2(t) was plotted against t as shown inFigure 12. The value of s2(t) is seen to approach an asymptote with increasing time.When the asymptote value of B(t) minus B(t) is plotted against time it decreases in a

complicated exponentlal-llke form as shown in Figure 13. Thus using the nonlineartheory of Nakada [10] it is concluded that the experimental B(t) for amalgam could bedescribed by the following equation:

00

O

which would Indeed describe the behavior of the curves seen in Figure 12 and Figure 13.It is therefore concluded that the combination elastic behavior of dental amalgam increep under the test conditions used can be described by means of the following equa-tion from viscoelastic theory [10,15]:

A( t ) CT + B^( t

)

CO

CT + r J(t) (l-e ^ J d T +

2

B(y) (l-e d Y

(29)

(30)

and since the loaded portion of the creep curve for amalgam is also composed of viscousstrain as well as combination elastic strain then the strain on the loaded por-tion is composed of the sum of the two as follows:

€ =

Thus

€ = Jq CT + CT Jj(r)(l-e d T+

JB(y) (l-e d Y

2 V "^4.CT + K CT t

(31)

(32)

Therefore applying both linear and nonlinear viscoelastic theory to the experimentalbehavior of amalgam under the test conditions used, a general equation describing thecreep behavior of amalgam as given by equation (32) above is obtained.

Applying the nonlinear generalization of the Boltzmann superposition principle as

developed by Nakada [10], to the above creep equation for amalgam, the stress-straincurves for various stress rate conditions were calculated for amalgam from the creepdata and compared to the experimental stress-strain curves obtained under those condi-tions. It is seen in Figure 14 that good agreement is obtained between the calculatedand experimental stress strain curves. Thus, it is concluded that in the case ofdental amalgam the results of creep tests can be directly related to those of stress-strain tests by use of viscoelastic theory [10,15].

6. Conclusion

Dental amalgam exhibits three types of viscoelastic phenomena: (l) instantaneouselastic strain (2) retarded elastic strain (transient creep) and (3) viscous strain(steady state creep).

The instantaneous elastic strain is assumed to be proportional to the appliedstress but the methods used in this study did not provide an Independent determinationof instantaneous strain.

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The combination of elastic plus retarded strain is a nonlinear function of stresand can bp represented by an equation-v^of the following form:

€' *= Act + ct^ where A and B are functions of time but not of stress

The viscous strain rate is also a nonlinear function of stress and can be repre-sented by an equation of the form:

€*v Kct^ where K and m are constants of the material

A general equation describing the creep behavior of amalgam was obtained by theapplication of both linear and nonlinear viscoelastic theory to the experimental be-havior of amalgam under the test conditions used. By applying the nonlinear general!zatlon of the Boltzmann superposition principle to this equation the results of creeptests can be directly related to results of stress-strain tests of amalgam.

7.

References

1. Rodriguez, M, L. and Dickson, G. £• D* ^ss . 28, 44l (1962).

2. Smith, D. L., Caul, H. J. and Sweeney, W. T. 535677 (1956).

3. Herschel, W. and Bulkley, R., Kollold-Z ., 39, 291-300 (1926).

4. Kauzmann, W. Trans . Am . Inst . Mining Met . Engrs . 143, 59 (l94l).

5. Chalmers, B. Proc . Roy. Soc . 156-A, 427 (1936),

6. Tyte, L. C. Proc . Physics Soc . London 50, I50 (1938).

7. McKeown, J. Inst . Metals . 60, 20 (1937).

8. Cottrell, A. H. Dislocations and Plastic Flow in Crystals , p. 210Oxford University Press, N. Y. ( 1953 ) .

~

9. Leaderman, H. J. Polymer Scl . I6, 261 (1955).

10. Nakada, 0. J. Phys . Soc . Japan 15, 2280 (i960).

11. Boltzmann, L. Sitzber , Kgl . Akad . Wiss Wien . 70, 275 (187^)»

12. Bland, D. R. The Theory of Linear Viscoelasticity. Pergamon Press, N. Y. (i960).

13. Leaderman, H. Elastic and Creep Properties of Filamentous Materials and OtherHigh Polymers . Textile Foundation, Washington, D. C. ( 1943 )

.

14. Ryge, G., Dibkson, G. and Schoonover, I. C, £. D. Res . 28, 44l (1952).

15. Leaderman, H. Rheology. Vo 1 . 2, p-l4 Academic Press, Inc., N. Y. (1958).

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’j!'

-V : •/ ‘'hSc^

' '"vi’-'l''

'^''

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.

“*•

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• i.;Vt i*ijRlV#'iv.-. ?-'

'

-ii- vw-j’ft* 1;.•«•?' •ti.-.'- «5js|!

•Oi-i- =••'., v; •^-

.'

,^^fS<,.‘.j|*-

Z'':^ •

>T.v/ vi’v.'; ;V^»' ^^i'’’*''

^i..I^'^^' t^v ,'^! j.’\;, n: •..V!. ' '

,,v;^,

^J - 'Lit ;;

'>' >' *;-'!' V

'

. •li :.J 'T<j a^RS-V'c '- .^' •

r^»- '’>.iyf!.^' ,<’ -/, V4 . y'> ,X, •,;•!

•.

'^.' : ) '

"Vffo '.vi

IC'^AW X.1V..i<

.

• .-u '>iW:'-X :!' 'j;:;.

;

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.

... * r'« ^

>—•••, ^.. ; V,. .V. iPii -/.

.

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--'i -*

< :"*/<& ' "-> 'VJ'' •••v Vw»>r* - : ^

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•••,y.,

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' '''.'A'-'":/'

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; "-",3s{'5""f '4- J^-*''.’*'^

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.

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TABLE 1

VISCOUS STRAIN RATES

Specimen - Run StressPS I

Strain perLoaded*X 10-5

MinuteUnloaded**

X 10-5

41-40-1 3807 5.6531 6.3511***41 -40-2 3807 5.9759 5.803739 -40-1 3731 4.9184 5

,

7'

449 *-k-*

39-40-2 3731 5.2273 5.082240-40-1 3731 7.4476 8.0372***43 -35-1 3458 3.1200 3.7086***43 -35-2 3458 3.2339 3.276243-35-3 3458 3.4171 3.438944-30-1 2959 1.9183 1.9778***44-30-2 2959 1.6821 1.676944-30-3 2959 1.7387 1.718541-30-3 2855 2.8429 2.955544-25-4 2465 0.9067 0.886735-25-1 2375 1,0521 1.3312***35-25-2 2375 0.9762 1.062135 -25-3 2375 1.0288 1.080338-25-1 2366 1.0909 1.3967***38-25-2 2366 •0,7832 0.821138-25-3 2366 0.7846 0.806135-20-4 1900 -0.3843 —35 -20-5 1900 0,4192 0.434838-20-4 1892 0.3487 0.358135-15-6 1425 0.1109 0.113338-15-5 1419 0.0761 0.076835-10-7 950 0.0306 0.033838-10-6 946 0.0232 0.024838-10-7 946 0.0170 0.017835-05-8 475 0.0083 0.0066

From slope of straight portion of loaded creep curve

From strain remaining in specimen after unloading and recovery

*** These values include effects of strain hardening and were not used incalculating the relation between stress and viscous strain rate.

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RECOVERY

OF

ELASTIC

(INSTANTANEOUS

PLUS

RETARDED

)

STRAIN

•H

ft in o o rH X rH -=t X on on o o OJ1 00 on X- OJ OJ ft =d* OJ -=t o VD LA o -=t

rH o o o\ m on o X ft OJ CO on X G\ OJ =t LAr—

I

liAt>- t-- 00 m o o rH on on -=t Lft LAon X rH rH rH rH 1

—1 rH rH rH rH rH rH

t{01

ft i-H iH rH (ft (ft CO -=t rH CO ft ft O) LA rH LA1 o o o (ft (ft OJ o (O (ft (ft o on ft on

00 o on rH Vft OJ CO OJ OJ (O ft rH (ft X 00 orH » • * * • • • • • • • • • • •

-=t VO [>- 00 00 (ft o rH rH rH OJ (M OJ OJ onm X 1

—1 rH rH rH rH rH rH rH rH

•HCO

ft -=t rH rH rH o on (O ft (ft X (D X o on 00 (ft

I 00 on on -=t X o on o o LA LA -=d* rH OJ Xon o <M 00 •=t tx o LA OJ rH on (ft CO rH -=t -=t -=t

on rH • • • • * • • • • • • • • •

CT\ 1ft Lft UD (ft X 00 ft ft ft ft o o o oOJ X rH rH rH rH

CO• • ft -=t ON O o on X 00 o LT( Lft o LA 00 OJ+> 1 OJ 00 vn (ft 1ft (ft ft (ft o Lft (ft rH ft 00cd -=i* o rH o (AJ LA ft -=t X o cv on =t u\

CO rH • • • • • » • • • • • • • •

* on on LA Lft Lft LA (O (D X X X X X Xc OJ X•HCd

L-PCO

•HCO

ft OJ OJ in on on on I—

1

OJ OJ 00 LA OJ (ft (J\ OJVft rH o\ 'vD lft (ft X (ft (ft OJ ft (O ft OJ X

o o OJ (ft X- ft (ft a\ Oi OI on on -=t -=J*

o\ 1—

1

• » • • « • • • • • • • •

CO on on on on on on LA LA LA LA Lft LA1

—1 X

•HCO oft on on X- X- on LA ft 1—

1

X o OJI rH VD (ft Cft (ft rH (ft <J\ OJ Lft 00 on (D CO 1

—(

OJ o rH 1

—1 OJ LA tx ft o 1

—1 -=t (O X 00 CO ft 1—

1

OJ- rH « • • « • « • • • • • •

OJ OJ OJ OJ OJ OJ on on on on on on on onrH X

CO =:t o t—

I

rH on (ft o LA CO rH LA o ft on 00ft 1 o o o o on LA X (O X 00 o 1

—1 OJ -=t LA

o on on on on on (D CT\ a^ ft o o o o o1—

1

1—

1

1

—1 1

—1 1

—I 1

—i 1

—1

11 1

—1 1

—1 rH OJ OJ OJ OJ OJ

o\ X

•HCO un in LA LA LA (O X X (ft LA LA LA LA LAa 1 -=t -=t -=t =t a) (O -=f -=t

o on (ft o o C5 o rH OJ on -=t

m 1

—I

i>- o o o I—

1 1

—J I 1 1

—1 1

—1 1

—1 1

—I rH 1

—I rH 1

(

rH

X

0) • LTV o o o o O o o o o o o o o o6 c OJ un o o o o o o o o o o o o o^ s o o I—

1

OJ on LA o o o o o o o o orH OJ on -=t lft X 00 ft

Each

value

Is

an

average

of

2to

7

determinations

of

the

difference

between

strain

recorded

at

the

time

Indicated

and

strain

recorded

Immediately

prior

to

removal

of

the

tensile

stress.

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TABLE 3

VALUES OF A(t) AND B^(t) IN EQUATION

A(t) a + B2(t)

Time A(t) B^(t)

Min. X lO"'^ X 10"^^

0.25 0.922 2.7070.50 1.122 2.8731.00 1.303 2,8282.00 1.309 3.3673.00 1.280 3.9045.00 1.322 4.284

10.00 1.363 4.98620.00 1.383 5.80430.00 1.409 6.11040.00 1.5^4 6.08250.00 1.589 6.19060.00 1.581 6.45470.00 1.570 6.67880.00 1.588 6.728

is combination of instantaneous and retarded elasticstrain

cr is stress

A(t) and B^(t) are constants for any time and are functionsof time but not of stress.

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^"\AAA^

\/W <

Figure

1.

Models

representing

types

of

viscoelastic

behavior:

A-spi’ing-elastic

behavior,

B-

dashpot

-

viscous

be-

havior,

C-Voigt

element

-

retarded

elastic

behavior

Page 20: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

0.75

o!c

I-

od

Figure

2.

Dimensions

of

the

dumbbell-shaped

tensile

specimen

of

amalgam.

Page 21: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

Figure 3. Tensile specimen in position for load application withoptical strain gages mounted on opposite sides of thespecimen.

Page 22: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

X NlVdlS

Figure

4.

Creep

and

recovery

of

amalgam

loaded

in

tension.

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in

00

Ood

m

O

in

iO

wCQ

(D CO "O dC 0 0 0

-P 0 d X5 PK! "H 1

1

cO PP cO 0 cO

TD nj > Pd d d dOJ D'TD 3 (D

CD 0 s(U TD ~>P-P iPl cO 0 0

b CO 0 0 > 0d XI d a

CO d CO

0) d -p 0

0 •H «HCO <U

• dCO ap

0 UCO

p 00 0 bO

_J CO 0uodd

0 pCO CO 0 d

•» d d x: ip -H

cn 0 D'-P 0 CO

0 CO Scn CO 1 0 d 0

X> -P 0 dLlI > CO p •

cO b P dor1

d 0 d P d0 M bO 0 CO 0

r— 0 0 a d >CO ^ 0 P P 0

•P d P CO 00 cO s x: ^ 0

U. X>CO

,

M s+ p 0

d

0 CO 0 cO d TOa d d dH d•H *H P CO

a>X« iH bO CO 0CO 0 d bO

iD d P rH S CO d0 X! 11 0 p0 P bC1 >d CO T3

_J P P’VL' dn 0CO CO 3

cO

0I—i d bO CD fH »H0 P 0 U d00 CO tH cd d

m0ddbOPPd

3 iriNm d 3 d Nivdis snoosiA 30 001A _ 0 _ _ _3 901

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o.oo to u> CVJ

55 (O fO 00 <T> CSJ

fO 00 5; 0)fO fO cvj OJ

K » o o 4 0 o0>

M

M

+

4

0

B

CO

CO

CD

u•pCO

c

o

**> o

4 0

4 D

4 D

o

CD •

I—I CQ

•H dca 3C d0)

-P I>-

<u oO P>

O(0

(O

0) OJCQ

Cd

CD O

(D CD

d hOCO

U ^CD CD

P >cp CO

cO

d>5 CO

d0>oo0d

Oro

d•H0dp>CO

Cd

0d3bO•iH

Ph

oX 3 A»3A003d NlVdlS

Each

curve

is

Page 25: BUREAU OF STANDARDS REPORT - NIST Page...NATIONALBUREAUOFSTANDARDSREPORT NBSPROJECT 311.05-20-3110560 June30,1964 NBSREPORT 8552 ProgressReport on ViscoelasticBehavior ofDentalAmalgam

(/>

Q.

b’

(ncnIxl

orI-(f)

9 ‘Ad3A003d NlVdlS

Figure

7.

Relationships

between

recovery

strain,

stress

and

time.

Plotted

points

are

averages

of

2to

7

deter-

minations.

Curves

are

least-squares

fits

of

the

equation

€'=

A(t)a

+

B2(t)cr2

to

the

data.

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to

Q.

b

COCOLlI

crh-co

"Oc I I

cfl QJ GX3 O

(B Om |>-

QJ CD CD

o ^ x:4^ 43

W •

C\J 43 CO•V d o 43d Cm CD CO

•H O CQ 43 TDcO CD -Hd ra d <di CDp CD a x:m hO CD CB 43

cO d CD

>5 d dod CD IB cO 43CD > CD d> co d aojO -H CB bO (D iH I

CD d 43 Pd co p ca^—

'

XI COC^J

d cB bo CD pqCD P P PCD d CO +^ P d >5p o p p bCD ftco ^p p pP

IB CD • d <ap CB P IIp p d CO'P O O P CLI

CB I—( P pd a p o do co oP d CB Pp . p p pCO CD s d co

I—I S d CO d

CD P CD P ap:: 43 P m CD

CO

CD

dd50pa

bX

o ofO

P 9oJ

*

‘ss3dis Aa aaaiAia AaaAooaa Nivais

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A(t)

Figure 9. Variation with time of the linear creep complianceterm, A(t), in the equation £'=k{t)a +

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(Ov-

TIME- MINUTES

Figure 10, Variation with time of (Aas~A(t)'^ where A(t) isthe linear creep compliance term' in the equation€'=Pi(t)a + B2(t)a2 and A^g is the asymptote valueof A(t) as t becomes large.

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A(t)

xio*^

1.6

T

l.l-

1.0

I

-1.0 1.0 2.0

log, t

3.0

Figure 11. Variation with log of time of the linear erepliance term^ in the equation f'=A{t)a

_1_4.0

com-f b2 ( t )a2

.

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qto

q q^ K)

(0,a

tnUJI-3

LxJ

Ph

qIT>

•gure

12.

Variation

with

time

of

nonlinear

creep

compliance

term

B2(t),

In

the

equation

^'^A(t)<y

+

B2(t)a2.

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CD

-P

C

(/)

UJHZ>Z

I

UJ

U)9- *"a

Figure

13.

Variation

with

time

of

(^Bag-B(t

)

J

where

B^(t)

is

nonlinear

creep

compliance

term

in

the

equation

€‘'=A(t)o-

+

B^(t)a^

anc3

B^g

Is

the

asymptote

value

of

B(t)

as

t

becomes

large.

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X

!sd-sS3dlS

Figure

l4.

Comparison

of

calculated

and

experimental

stress-

strain

curves.

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