Sinusoidal deformation model

The rheological behavior of a viscoelastic material can be investigated by applying a small-amplitude sinusoidal deformation. The behavior can be described by a mechanical model, called the Maxwell model [33], consisting of an elastic spring with the Hookean constant, G , and a dashpot with the viscosity, r/<,. The variation of storage modulus (G ) and loss modulus (G") with shear frequency, O), are given by the equations... 

Once the viscoelastic response of polymer materials is figured out qualitatively, one needs appropriate tools to develop a quantitative model. Firstly, one should bear in mind that the response described by the time-dependent modulus, as in any kind of response function for a casual, time-independent and linear system, can of course be reformulated in the frequency domain. Having, as a problem input, a sinusoidal deformation at a given angular frequency (cu), y(r) = Yo sincut, the corresponding stress, a(t), in the material will have a component in phase and a component in quadrature with respect to the input ... 

We have constructed a model which, unlike previous models of IC-AFM, takes the tilt of the cantilever into account. The model assumes Hertzian tip-sanple contact, with both in-plane and out-of-plane dissipative conqmnents (29). The key result is that con onents of motion bodi normal and parallel to the sanple occur, and therefore in-plane dissipative processes can cause phase shifts. Using parameters appropriate for oin system, we solve for die steady state motion of the tip. The model indicates a maximum tip-sanqile in-plane tip motion of =49.9 0.1 pm parallel to the sanqile. The distance S is extremely small, and it is difficult to make firm distinctions between friction and shear deformation at such a small scale, as discussed below. The inqiortant result is that is virtually independent of the in-plane damping. Furthermore, the model produces a nearly sinusoidal tip motion, indicating diat Eq. (2) remains valid for the tilted-cantilever geometry. 

We consider the system model depicted in Figure 5.2, which describes the self-cleaning mechanism through the following sub-models (1) droplet model, (2) substrate surface model, and (3) particle-droplet interaction model. Since quasi-static conditions are considered in this work, it is convenient to treat the droplet as a hydrostatic bulk and a deformable liquid membrane. The latter is modeled using the stabilized FE formulation introduced by Sauer [15], which captures the in-plane equilibrium of the membranes due to constant surface tension. The multi-scale nature of self-cleaning surfaces can be mathematically modeled as 2D sinusoidal functions as done by Bittoun and Marmur [19], and Iliev and Pesheva [20],... 

Thus in all our discussions for simple shear we must realize that the models can be applied to other types of deformation. In the next section we develop a general model for linear viscoelastic behavior in only one dimension. Then we extend it to three dimensions, and in Section 3.3 we examine its behavior for different deformation histories stress relaxation, creep, and sinusoidal oscillation. 

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