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The Quarterly Journal of Mechanics and Applied Mathematics 1954 7(4):399-409; doi:10.1093/qjmam/7.4.399
© 1954 by Oxford University Press
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DYNAMIC BEHAVIOUR OF LINEAR RHEOLOGICAL BODIES UNDER PERIODIC STRESSES

S. IRMAY {dagger}

( Israel Institute of Technology Haifa )

The simpler linear rheological bodies—such as the elastic and Kelvin's solids, Maxwell's and Newton's viscous fluids—obey a generalized linear differential equation with constant coefficients between the strain and stress deviators and their time rates. The various coefficients are correlated to the asymptotic rigidity or static elastic modulusG, elastic firmness or dynamic modulusH, solid viscosity µ, times of relaxationR and of lagging or retardationL, and the endosity {eta}.{ddagger}

Expressing the time and stress in non-dimensional forms, a universal equation is obtained, dependent on a single non-dimensional parameter, the ‘time factor’ {tau} = R/L = G/H. This defines a principle of similitude for all bodies of identical {tau}.

The mechanical and thermodynamic study of periodic, impulsive, and transient stresses leads towards a new classification of the linear bodies, based on the value of {tau}:

  1. The exothermal or dissipative bodies ({tau} < 1) are less rigid than firm, less strained than elastic solids, dissipate energy in a cycle. They warm up adiabatically. Here belong also Maxwell's elastico-viscous fluids and Kelvin's firmo-viscous solids (t = 0).
  2. The endows solids ({tau} > 1) are more rigid than firm, more strained than elastic solids, may show a' negative' dissipation in a cycle, which is compensated by internal structural changes. This is not in contradiction with the second law of thermodynamics and may explain the fatigue of such solids.
  3. The homothermal solids ({tau} = 1) are as rigid as firm, show no apparent dissipation in a cycle. A particular case is that of elastic solids.

The stress-strain diagram under periodio stresses tends towards an ellipse, which shows accommodation of the body. The smaller axis passes through a maximum for a critical frequency, and the body has two independent elastic moduli—a static and a dynamic one. These facts were experimentally confirmed for certain plastics.


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