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Constitutive Equations in Differential Operator Form

  • Danton Gutierrez-LeminiEmail author
Chapter

Abstract

The mechanical response of a viscoelastic material to external loads combines the characteristics of elastic and viscous behavior. On the other hand, as we know from experience, springs and dashpots are mechanical devices which exhibit purely elastic and purely viscous response, respectively. It is then natural to imagine that the equations that relate stresses to strains in a viscoelastic material could be represented with an appropriate combination of equations which relate stresses to strains in springs and dashpots. To develop this idea, Sect. 3.2 examines the response of the linear elastic spring and linear viscous dashpot to externally applied loads. The response equations for these simple mechanical elements are formalized in Sect. 3.3, with the introduction of so-called rheological operators. As it turns out, because combinations of springs and dashpots require the addition and multiplication of constant and first derivative operators, it turns out that the constitutive equation of general arrangements of springs and dashpots, such as are needed to reproduce observed viscoelastic behavior, must be represented by linear ordinary differential equations whose order depends on the number, type, and specific arrangement of the springs and dashpots. The physical significance of the coefficients in the resulting differential equations is examined also, and the proper form of the initial conditions established. As will be seen, the mere presence or absence of some of the coefficients of a differential equation reveals whether the particular arrangement of springs and dashpots it represents will model fluid or solid behavior, and whether it will exhibit instantaneous, elastic response. A general approach to establishing rheological models is presented in Sect. 3.4, and applied in Sects. 3.5 through 3.7 to develop the differential equations, and examine the behavior of simple and general rheological models.

Keywords

Compliance Creep Damper Dashpot Deviatoric Fluid Kelvin Laplace Maxwell Model Modulus Pressure Recovery Relaxation Spherical Spring Strain Stress Relaxation Retardation Solid Spectrum 

References

  1. 1.
    D.L. Kreider, R.G. Kuller, D.R. Ostberg, Elementary Differential Equations (Addison-Wesley, Reading, 1968), pp. 66–69Google Scholar
  2. 2.
    R.M. Christensen, Theory of Viscoelasticity, 2nd edn. (Dover, NY, 2003), pp. 14–20Google Scholar
  3. 3.
    A.D. Drosdov, Finite Elasticity and Viscoelasticity (World Scientific, Singapore, 1996), pp. 250–255Google Scholar
  4. 4.
    J.D. Ferry, Viscoelastic Properties of Polymers, 3rd edn. (Wiley, NY, 1980), pp. 60–67Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Special Products DivisionOil States Industries, IncArlingtonUSA

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