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Research Papers

A Theory Relating Creep and Relaxation for Linear Materials With Memory

[+] Author and Article Information
R. C. Koeller

 Mechanical Design Group, 960 Stonebridge Suite 11, Platteville, WI 53818-2071ralph_koeller@yahoo.com

If beta is both positive and negative, then the Able’s operator considered as a generalized functional becomes the fractional integral and fractional derivative (see Sec. 3).

J. Appl. Mech 77(3), 031008 (Feb 08, 2010) (9 pages) doi:10.1115/1.4000415 History: Received August 31, 2008; Revised August 27, 2009; Published February 08, 2010; Online February 08, 2010

The purpose of this paper is to suggest a linear theory of materials with memory, which gives a description for the similarities resulting when the various analytical and experimental methods used to reduce the creep and relaxation data are imposed on the observational changes in curvature that take place in both the creep compliance and relaxation modulus graphs. On a Log-Log graph both have one, two, or at most three pairs of changes in curvature depending on whether the material is a fluid or solid. These changes in curvature have been observed in many experiments and various regions have been discussed and classified. Section 1 gives a few of the many applications of fractional calculus to physical problems. In Sec. 2 an equation that contains both integration and differentiation is presented using geometrical observations about the relationship between the changes in curvature in the relaxation modulus and creep compliance based on published experiments. In Sec. 3 the generalized function approach to fractional calculus is given. In Sec. 4 a mechanical model is discussed. This model is able to share experimental data between the creep and relaxation functions, as well as the real and imaginary parts of the complex compliance or the complex modulus. This theory shares information among these three experimental methods into a unifying theory for solid materials when the loads are within the linear range. Under a limiting case, this theory can account for flow so that the material need not return to its original shape after the load is removed. The theory contains one physical parameter, which is related to the speed of sound and a group of phenomenological parameters that are functions of temperature and the composition of the material. These phenomenological parameters are relaxation times and creep times. This theory differs from the classical polynomial constitutive equations for linear viscoelasticity. It is a special case of Rabotnov’s equations and Torvik and Bagley’s fractional calculus polynomial equations, but it imposes symmetry conditions on the stress and strain when the material is a solid. Sections 5,6 are comments and conclusions, respectively. No experimental results are given at this time since this paper presents the foundations of materials with memory as related to experimental data. The introduction of experimental data to fit this theory will result in the breakdown of an important part of this research.

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Grahic Jump Location
Figure 1

The mechanical model for a third order transition

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