Laplace transform, linear viscoelasticity

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives... 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

The linear formalism implies, that for many classes of problems in deformation, solutions in the viscoelastic case can be simply obtained from elastic solutions by Laplace transform methods as for isotropic materials. Further in particularly simple cases this reduces to simple replacement of constants in elastic solutions by their time dependent analogues in creep. Most of the work of determining significant creep parameters for oriented materials falls into this latter class. 

Derivation of viscoelastic beam deflection equation in the time domain It is instructive to derive the deflection equation for a viscoelastic beam without resorting to Laplace transforms. Consider the undeformed and deformed beam shown in Fig. 8.5. Making the assumptions (the same as in elementary solid mechanics) of small deformations, linear behavior, and a non-warping cross-sections (plane sections remain plane) will give the relations. 

Obviously, the above transformed governing equations for a linear viscoelastic material (Eqs. 9.33- 9.36) are of the same form as the governing equations for a linear elastic material (Eqs. 9.25 - 9.28) except they are in the transform domain. This observation leads to the correspondence principle for three dimensional stress analysis For a given a viscoelastic boundary value problem, replace all time dependent variables in all the governing equations by their Laplace transform and replace all material properties by s times their Laplace transform (recall, e.g., G (s) = sG(s)),... 

The relaxation time spectrum can be calculated exactly from the measured stress relaxation modulus using Fourier or Laplace transform methods, and similar eonsiderations apply to the retardation time spectrum and the creep compliance. It is more convenient to consider these transformations at a later stage, when the final representation of linear viscoelasticity, that of the complex modulus and complex compliance, has been discussed. 

A complete exposition of the mathematical structure of linear viscoelasticity has been given by Gross [7]. Here we will only summarise certain parts of his argument to illustrate the use of Laplace and Fourier transforms in establishing the formal connections between various viscoelastic functions ... 

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