Mechanical elastic response modeling

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

As the term implies, viscoelasticity is the response of a material to an applied stress that has both a viscous and an elastic component. In addition to a recoverable elastic response to an applied force, polymers can undergo permanent deformation at high strains, just as was the case for metals and some glasses, as described previously. The mechanism of permanent deformation is different in polymers, however, and can resemble liquid-like, or viscous flow, just like we described in Chapter 4. Let us first develop two important theoretical models to describe viscoelasticity, then describe how certain polymers exhibit this important property. 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

Molecular modeling techniques have been used to predict and interpret mechanical properties of polymers [88-95]. Theodorou and Suter [88, 89] found that the internal energy contribution to the elastic response is much more important than the entropic contribution for glassy polymers by a thermodynamic... 

Furthermore, yet to be computed by any program is the fundamental thermo-mechanical transduction wherein the cross-linked elastic-contractile model proteins contract and perform mechanical work on raising the temperature through their respective inverse temperature transitions. These results first appeared in the literature in 1986 and have repeatedly appeared since that time with different preparations, compositions, and experimental characterizations. Additionally, the set of energies converted by moving the temperature of the inverse temperature transition (as the result of input energies for which the elastic-contractile model protein has been designed to be responsive) have yet to be described by computations routinely used to explain protein structure and function. 

The simplest mechanical model, the hookean spring element, has an elastic response. The spring is an energy storage element. It releases its energy when it returns to its original form. When subjected to an instantaneous stress Oo, the spring has a response with a strain Eq [10] ... 

The sample-tip-cantilever system can be modeled as a mechanical system with springs and dash-pots 11,12). Solving the motion equations of this model at low frequency (i.e. below the cantilever resonance frequency) and neglecting the damping constants (i.e. neglecting viscoelastic effects in polymers) leads to the following relation for the ratio between the sample modulation amplitude, z, and the tip response amplitude, also called the dynamic elastic response ... 

Consider, for example, the creep response of the four-parameter model (Fig. 18.8). For this model, a logical choice for A would be the time constant for its Voigt-Kelvin component, Jja/Gz- For De> 1 (t - Ac), the Voigt-Kelvin element and dashpot 1 will be essentially immobile, and the response will be due almost entirely to spring 1, that is, almost purely elastic. For De 0 (t, > A ), the instantaneous and retarded elastic response mechanisms have long since reached equilibrium, so the only remaining response will be the purely viscous flow of dashpot 1, and the deformation due to viscous flow will completely overshadow that due to the elastic response mechanisms (imagine the creep... 

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