Model-dependent strain

It is worthwhile to re-emphasize briefly how model-dependent strain is at this point. If we employ the value for the ethyl stabilization for fluoroethylene (Table 2) we... 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

FIGURE 30.9 Modeling the strain dependence of complex modulus. 

We observe that, in addition to the strain rate a, the computational model depends on the equivalence ratio (j) which is a measure of the relative proportion... 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Here is the element viscosity, is the element modulus, and Ay is the relaxation time. If this model material is placed in a constant stress environment for a fixed time and then the stress is removed, then the time-dependent strain will have the recovery characteristics shown in Fig. 3.11. 

Figure 5.60 (a) Maxwell spring and dashpot in series model of viscoelasticity and (b) constant stress conditions resnlting in time-dependent strain. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc. 

Notice that the compliance is inversely proportional to the modulus. Equations for the time-dependent strain can be developed for the four-element model shown above, or for any combination of elements that provide a useful model. The corresponding time-dependent compliance can then be determined using Eq. (5.75), or the time-dependent modulus using an analogous equation. 

The analysis has shown that PAI may only be negative, and PAB ( both positive and negative. Therefore, the thermal effect accompanying a reversible stretching of the model depends on the ratio between p and PA,n and may be a function of strain even at small strains. Besides, Poisson s ratio for such a heterogeneous model may exeed 0.5, Direct measurements of Poisson s ratio for a number of various oriented crystalline polymers are consistent with this suggestion (see Table 5). 

Here is another instance of the extreme model dependency of strain energies in the fluorocarbon series. A strainless C(F)2(C)2 group increment of - 104.9 kcal mol" had earlier been derived ". In principle, one might expect that this increment would be applicable to calculating the strain energy of hexafluorocyclopropane. Applying this value, we concluded that the total strain in the molecule was 80.9 kcal mol" This was even higher than the value calculated by Bernett some years earlier 68.6 kcal mol" We were buoyed by the fact that the endothermicity of equation 18 indicated a strain of... 

The first four terms of the function are commonly found in molecular mechanics strain energy functions, and they are modified Hooke s law functions. The last term has been added to insure the proper stereochemistry about asymmetric atoms. A model is refined by minimizing the highly nonlinear strain energy function with respect to the atomic coordinates. An adaptive pattern search routine is used for the strain energy minimization because it does not require analytical derivatives. The time necessary to obtain good molecular models depends on the number of atoms in the molecule, the flexibility of the structure, and the quality of the starting model. 

For viscoelastic liquids, the Maxwell model can be used to qualitatively understand the stress relaxation modulus. In the step strain experiment, the total strain 7 is constant and Eqs (7.101)-(7.103) can be combined to give a first order differential equation for the time-dependent strain in the viscous element ... 

Peeling [Dembo et al., 1988] Clamped elastic membrane Bond stress and chemical rate constants are related to bond strain Bonds are linear springs fixed in the plane of the membrane Chemical reaction of bond formation and breakage is reversible Diffusion of adhesion molecules is negligible Critical tension to overcome the tendency of the membrane to spread over the surface can be calculated Predictions of model depend whether the bonds are catch-bonds or slip-bonds If adhesion is mediated by catch-bonds, then no matter how much tension is applied, it is impossible to separate membrane and surface... 

A natural extension of linear elasticity is hyperelasticity (Ogden 1997). Hyperelasticity is a collective term for a family of models that all have a strain energy density that depends only on tiie currently applied deformation state (and not on the history of deformations). This class of material models is characterized by a nonlinear elastic response, and does not capture yielding, viscoplasticity, or time dependence. Strain energy density is the energy that is stored in the material as it is deformed, and is typically represented either in terms of invariants of the deformation gradient (F) /i, I2, and /, where... 

Creep. Creep is a time-dependent strain increase under a constant stress. As already mentioned, the constant stress can be quite simply provided by a gravitational field of the earth. The creep behavior is most often analyzed in terms of the Kelvin-Voigt model in which a spring and a dashpot are parallel. The model is characterized by a constant representing the elastic (modulus) and viscous flow (viscosity) deformations. From the geometry of model, individual strain in each element is equal to total strain and applied stress is supported jointly by the spring and dashpot. 

Issues of Material Compressibility. There is a full theory of compressible and nonlinear viscoelastic materials that would be equivalent to the compressible finite deformation elasticity theory described above (eq. 39), but more complicated because of the need to develop an expansion of the time-dependent strain potential function as a series of multiple integrals (108,109). One such formahsm is discussed briefiy under Lustig, Shay and Caruthers Model. Here a simphfied model that is based upon the K-BKZ framework with a VL-like kernel function (98) is examined. 

For depth-sensing nanoindentation, a controlled, variable force is applied to a sample by the indenter and the resulting displacement of the indenter is measured. The resulting load vs. displacement data, together with the indenter geometry, can be analyzed to obtain hardness and elastic modulus using well established mechanical models (14). The simultaneous measurement of load and displacement also allows study of creep (time dependent strain response due to a step change in stress) (15,16). 

However, amorphous water-soluble materials, such as food materials, deform viscoelastically. The deformation and relaxation behavior of such materials can be described by means of various viscoelastic models. Depending on the nature of the stress/strain applied, either the storage and loss modulus or the elasticity and the viscosity are included as material parameters in these models. These rheological material parameters depend on the temperature and the water content as well as on the applied strain rate. The viscoelastic deformation enlarges the contact area and decreases the distance between the particles (see Fig. 7.3). If the stress decreases once again, the achieved deformation is partially reversed (structural relaxation). 

The elastic behaviour of the partially polymerized lattice is model dependent. The two simplest cases are vhen (a) the two con nents are subjected to equal strain and (b) the two con nents are subjected to equal stress (22). Case (a), the Voight limit, corresponds to con nents coupled in pairallel, i.e. long polymer chains, and (b), the Reuss limit, corresponds to con nents coupled in series, i.e. short polymer chains. Baughmann (20) used case (a) to calculate the lattice strain. 

In other words, the QLV model reflects strain history dependent stress and fading memory. In order to express relaxation properties, Prony s series (97) might be used, i.e.,... 

Note that this definition does not subtract the instantaneous or elastic strain to obtain creep. This is due to the difficulty in separating the two components of the time-dependent strain required by the Kelvin or Voigt model of viscoelasticity. 

Knowledge of the rheological properties of food pastes, slurries and sauces, such as ketchup, mayonnaise and salad creams, is important both for quality assurance and for optimizing industrial flow and mixing processes. Unfortunately, many food slurries and pastes are opaque and do not lend themselves to flow studies with conventional techniques such as laser Doppler anemometry. Moreover, conventional rheological measurements are model-dependent in that it is necessary to fit the data by assuming a function relationship between the stress and strain (or strain rate) and to assume a set of boundary conditions (such as slip or stick) at the fluid-container... 

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