Thermodynamics, stability and non-linear oscillations of viscoelastic solids — I. Differential type solids of second grade

• Published in: International journal of non-linear mechanics 1996, Vol.31 (4), p.495-516
• Main Authors: Fosdick, Roger L., Yu, Jang-Horng
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; stability; Structural and continuum mechanics; thermodynamics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves; viscoelastic solids
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/0020-7462(96)00005-4

We study the thermodynamics and stability of a viscoelastic second grade solid whose action is characterized by two microstructural coefficients α 1 and α 2 in addition to the Newtonian viscosity μ. We show that it is both necessary and sufficient that μ ⩾ 0, α 1 ⩾ 0 and α 1 + α 2 = 0 if the material model is to be compatible with thermodynamics and its free energy is to be at a local minimum in equilibrium. Then, we construct a stability theorem for second grade solids which undergo mechanically isolated motions wherein it is shown that the motion of the body relative to its center of mass will dissipate away in time. The stability theorem is exemplified by investigating the free oscillation of cylindrical and spherical shells where the equilibrium state is globally stable. When μ = 0, but α 1 ≠ 0, the shells exhibit a larger period than if they were purely elastic in the classical sense.