Stress-strain ensembles

B. Implementation of stress-strain ensembles for open and closed systems... 

Stress-Strain Ensembles for Open and Closed Systems... 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

Najafbadi and Yip (18) have investigated the stress-strain relationship in iron under uniaxial loading by means of a MC simulation in the isostress isothermal ensemble. At finite temperatures, a reversible b.c.c. to f.c.c. transformation occurs with hysteresis. They found that the transformation takes place by the Bain mechanism and is accompanied by sudden and uniform changes in local strain. The critical values of stress required to transform from the b.c.c. to the f.c.c. structure or vice versa are lower than those obtained from static calculations. Parrinello and Rahman (14) investigated the behavior of a single crystal of Ni under uniform uniaxial compressive and tensile loads and found that for uniaxial tensile loads less than a critical value, the f.c.c. Ni crystal expanded along the axis of stress reversibly. 

The constant-temperature, constant-stress ensemble (NST) is an extension of the constant-pressure ensemble. In addition to the hydrostatic pressure that is applied isotropically, constant-stress ensemble allows you to control the xx, yy, zz, xy, yz, and zx components of the stress tensor (sometimes also known as the pressure tensor). This ensemble is particularly useful if one wants to study the stress-strain relationship in polymeric or metallic materials. 

Fig. 8.8 An ensemble-average extensional equivalent stress-strain curve of amorphous polypropylene, derived from axial extension and shear-flow ensembles of separate simulations. The smoothed broken line, average curve, drawn-in by eye, shows clear elasto-plastic behavior and a beginning yield phenomenon (from Mott et al. (1993) courtesy of Taylor and Francis).

Figure 8.10 gives an ensemble-average uniaxial stress-strain curve for PC and the associated system pressure resulting from this monotonic deformation. The smoothed stress-strain curve shows many of the same special features as the two individual stress-strain curves of PP in Fig. 8.8, as well as the ensemble-average result of Fig. 8.8. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

The term grand mixed isostress iiostrain ensemble is used to indicate that the slit-pore is materially coupled to its environment and that its thermodynamic state depends on the control of a set of stresses and strains. 

As we showed in Section 2.5.4, thermal averages in isostress isostrain ensembles can be related through a Laplace transformation. Hence, for the conjugate stress r and strain A, wc may employ Echange variables according to Tja — t and —> A giving... 

Fig 8.10 An ensemble-average shear-stress-shear-strain curve of 12 pure shear simulations of PC (top part), for a simulation method similar to the simulations for polypropylene of Fig. 8.8. The lower part shows the ensemble-average system pressure associated with the shear behavior (from Hutnik et al. (1993) courtesy of the ACS). 

More recently, stress and strain have been incorporated as a conjugated pair of slow dynamic variables to extend the model of the SCF theory. This allows us to capture some effects of viscoelasticity [76]. Similar to the evaluation of the single chain partition function by enumeration of explicit chain conformations, one can simulate an ensemble of mutually non-interacting chains exposed to the effective, self-consistent fields, U and W, in order to... 

Results are presented here from averaging over about 100 NTLxOyya z MD trajectories for each stress relaxation experiment, initiated at ensembles of strained configurations of two PE melt systems a 32-chain C24 and a 40-chain Cvg PE melt. 

The issue of dislocation formation in a strained epitaxial heterostructure was the focus of attention in the preceding chapter. Residual stress was assumed to originate from the combination of a mismatch in lattice parameters between the materials involved and the constraint of epitaxy. The discussion in Chapter 6 led to results in the form of minimal conditions which must be met by a material system, represented by a geometrical configuration and material parameters, for dislocation formation to be possible. Once the values of system parameters are beyond the point of fulfilling such minimal conditions, dislocations begin to form, propagate and interact. The ensemble behavior is usually termed strain relaxation. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

In these relations angular brackets denote volume averages of the Cauchy stress tensor and the small strain tensor, respectively. For the correct use of volmne and ensemble averages in connection with random and periodic microstructures the reader should consult [Torquato... 

Several nanotube-polymer composite systems have been extensively studied by means of Raman spectroscopy under strain ranging from —1 to 4%. For more detailed study on this topic, refer to Ref [81]. It is important to stress that in these measurements, a large number of nanotubes are measured simultaneously resulting in a Raman response that is an ensemble average of many different nanotube diameters and chiralities. Figure 10.7 shows the strain-induced shift of—15 cm /%... 

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