Theory of Non-linear Responses

Viscoelastic behavior of dental amalgam:

“…Thus from Figure 6 it was concluded that the retarded elastic strain as a function of time could be described by means of conventional viscoelastic theory assuming either a finite or infinite number of Voigt elements while the combination elastic strain being a nonlinear function of stress indicated the need for the use of nonlinear viscoelastic theory as developed first by Leaderman[9l and later more generally by Nakada [10]. Therefore, 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 Agg is the asymptote value of A(t) as t becomes very large.…”

“…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 nonlinear theory of Nakada [10] it is concluded that the experimental B(t) for amalgam could be described 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 in creep under the test conditions used can be described by means of the following equation from viscoelastic theory [10,15]: Applying the nonlinear generalization of the Boltzmann superposition principle as developed by Nakada [10], to the above creep equation for amalgam, the stress-strain curves for various stress rate conditions were calculated for amalgam from the creep data and compared to the experimental stress-strain curves obtained under those conditions.…”

“…In many materials this relation has been found to be nonlinear. The mathematical description of the nonlinear case has been treated In various publications [9,10] …”

“…Is the general creep compliance For example, in the linear creep behavior as seen in equation (10) , the general creep compliance J(t) is of the form: (14) In the case that creep strain Is linearly related to the applied stress, the stress-strain behavior for a material may be related to the creep behavior by means of the linear superposition principle developed by Boltzmann [ 11 ] as shown below Is the Increment of stress from time 9 to time ( 9 + d 9 )…”

“…For example, applying the Boltzmann superposition principle to the example of creep given In equation (10) gives the following relation between creep behavior and the strain measured In a stress-strain test [12] [10] may be used to relate nonlinear retarded elastic creep behavior to the stress-strain behavior of the material as given: …”