+

σ(T)

longer intervals until the strain rate became constant (usually after approximately 1.5 hr). At the end of this period, the load was removed, a strain reading was taken immediately and the recovery curve was followed by reading first at 15 s intervals and then at longer intervals until the strain became constant. Strain readings obtained on the two sides of the specimen were averaged and strain was 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 in. Since a gage length of 0.25 in was used this is equivalent to a strain of 8 × 10-5. Thus, in the results given below, differences in strain of 1 x 10-4 are approximately equal to the minimum reading difference.

Loads placed on the specimen (with a nominal 0.01 in2 cross-sectional area) varied from 5 to 40 lb giving stresses from approximately 500 to 4,000 psi. Most specimens were used for several runs, first at high and then at lower stresses. The first loaded creep run was considered a strain hardening treatment and data obtained on these runs were not used in the calculation of results other than for viscous strain rate. All runs were made at 23±1 °C.

FIGURE 1. Dimensions of the dumbbell-shaped tensile specimen of

amalgam.

FIGURE 2. Tensile specimen in position for load application with optical strain gages mounted on opposite sides of the specimen.

number of different stress levels in figure 3 indicate that at room temperature amalgam exhibits three different types of viscoelastic phenomena: (1) 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 curve by taking the slope of the straight line portion of the curve, and was also determined from the recovery portion of the creep curve by dividing the value of the recovery strain (the permanent strain in the specimen) by the total time the load was on the specimen. The viscous strain rates for any given creep curve as calculated from the loaded and recovery portions of the curve were found to agree fairly well as shown in table 1. The log of the viscous strain rate was found to be a linear function

The value of m for amalgam is the value of the slope of the curve in figure 4, while the value of k is the antilog of the viscous strain rate value at a value of applied stress σ of 1 psi. For the dental amalgam used in this investigation values for K of 2.85 × 10-19 and 4.98 × 10-1 19 were obtained from loaded and unloaded data respectively, and values of 3.99 and 3.92 were obtained for m.

The strain developed in amalgam due to the other two phenomena (1) instantaneous elastic strain and (2) retarded strain can be determined from the strain recovery since 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 has been unloaded (that is in the recovery portion of the creep curve) are subtracted from the strain value on the creep curve at the instant just before unloading of the specimen. This difference is plotted against recovery time t = Ti-Tu where T is the time at which the specimen was unloaded and

TABLE 1. Viscous strain rates

Tis the time of the strain value on the recovery portion of the curve. These difference values, e', are seen plotted against the recovery time for each load or stress in figure 5. These plots are a measure of the combination of the elastic and retarded elastic strain behavior of dental amalgam as a function of time for various stress levels. A measure of the combination of elastic and retarded strain may also be obtained from the loaded portion of the creep curve by taking values off the loaded creep curve and subtracting the viscous strain accumulated in the specimen at that time. The accumulated viscous strain at any time may be calculated by multiplying the 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 portion and the viscous strain value at a corresponding time is a measure of the combination of the instantaneous and retarded elastic strain. However, a small error in the viscous strain rate causes a large error in the difference value. Therefore, the plot of the combination of elastic and retarded elastic strain versus time as obtained from the loaded creep curve is subject to large possible

error.

Using the approximation of Nakada's general formulation described by Leaderman, McCrackin, and Nakada [7], the ordinates of the inverted recovery curves as described above should be greater in the early portion of the time scale than the ordinates of the loaded portion of the creep curves minus the viscous strain. The difference between the curves should rise to a maximum within the first few minutes and gradually decrease to zero as time increases. However, as shown in figure 6, the loaded curves appear to lie above the recovery curves. Fitting such data to

Nakada's formulation would be very difficult. Moreover, the differences are not large and the method of obtaining values for the loaded curves is subject to considerable error because the loaded curves had to be corrected by subtracting the viscous component. Thus it is believed that the differences between the curves are not significant and so the curves were treated as though they are superimposable. Therefore, the data reported for the combination of elastic and retarded elastic strain were obtained from the recovery portion of the creep curves.

The combination elastic strain (instantaneous and retarded elastic strain) becomes asymptotic with time in accordance with theory as seen in figure 5. The combination elastic strain values were plotted as a function of the various stress levels for corresponding time, as shown in figure 7. The combination elastic strain is seen to be a nonlinear function of the applied stress. When the combination strain values were divided by their corresponding stresses and then plotted against the corresponding stress for a fixed time, a linear plot was obtained for each fixed time as illustrated in figure 8; this result indicated that the combination elastic behavior of amalgam as a function of applied stress under the test conditions could be represented by an equation of the form: e' A(t)o+B2 (t)σ2,

where

(13)

e' is the combination of elastic and retarded elastic strain, and

σ is the applied stress.

O

2

STRESS, σ, psi.

FIGURE 7. Relationships between recovery strain, stress and time.

Plotted points are averages of 2 to 7 determinations. Curves are calculated from leastsquares fits of the equation e'lo=A\n}+B^\nơ to the data with error assumed to be in €/σ only.

1.0

2

A(t) and B2(t) are functions of time but not of stress. The value of A(t) for any time value is the intercept at σ=0 of the plot for that time value as shown in figure 8 while B2(t) is the slope of the straight line for that time value. It is also noted in figure 8 that as a function of stress the 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 same functional relation to the stress. Since the recovery curves were found to be superimposable upon the loading curves minus the viscous portion, it appeared that the nonlinear material did not obey the approximation of nonlinear generalization previously noted [7]. It was then questioned whether the retarded behavior of amalgam was truly nonlinear or whether the nonlinear behavior could be attributed to the geometry of the specimen being observed. However, longer specimens were tested and found to give the same result, also photoelastic specimens were made and the stress distribution over the area observed was found to be uniform. It is therefore concluded that the observed nonlinear behavior of dental amalgam is not a geometric artifact, but an intrinsic phenomenon in the material. The difference between the recovery curves and the loading curves minus the viscous portion is given by [7]:

The values for A(t) and B2(t) were determined by fitting curves to the data by the method of least squares and were tabulated as a function of time as shown in table 2.3 The A(t) values were plotted in figure 9. The A(t) values are seen to approach an as a function of the corresponding t values as shown asymptote, Aas as t→. Thus, the curve as shown in figure 9 could be represented by the following equation from linear viscoelastic theory since A(t) is the linear

term in stress:

(15)

FIGURE 9.

Variation with time of the linear creep compliance term, A(t), in the equation e' = A(t)σ+ B2(t)σ2.

1 The notation B(1) rather than B(t) is used for consistency with the form of the notation of Leaderman, McCrackin, and Nakada [7].

* Strain values for different stresses were obtained at each of the specific times listed in table 2 (and for many other times not listed in the table) by interpolation between the recorded values of strain along each of the 25 to 30 creep and recovery curves. Then the strain values for a specific time were fitted to the equation e'σA(t) + B2(t)σ to obtain the values of A(t) and B (1) for that specific time.