Failing microstructure model

Non-linear constitutive equations are developed for highly filled polymeric materials. These materials typically exhibit an irreversible stress softening called the "Mullins Effect." The development stems from attempting to mathematically model the failing microstructure of these composite materials in terms of a linear cumulative damage model. It is demonstrated that p order Lebesgue norms of the deformation history can be used to describe the state of damage in these materials and can also be used in the constitutive equations to characterize their time dependent response to strain distrubances. This method of analysis produces time dependent constitutive equations, yet they need not contain any internal viscosity contributions. This theory is applied to experimental data and shown to yield accurate stress predictions for a variety of strain inputs. Included in the development are analysis methods for proportional stress boundary valued problems for special cases of the non-linear constitutive equation. 

In contrast to the nano-scale, where the periodic arrangement of atoms on crystal lattices is well established, and the macro-scale, where a continuous distribution of matter is assumed, adequate quantitative descriptions are notably lacking for structure at the micro- and mesoscales, where properties are described in terms of the behavior of dislocations, material in grains, particles of different phases and the boundaries among them. The traditional means of describing these microstructural attributes with descriptive terms that call to mind familiar shapes fails to provide an adequate quantitative basis for transferring this information to quantitative models. 

Fracture Stress and Strain. Yielding and plastic deformation in the schematic representation of tensile deformation were associated with microfibrillation at the interface and stretching of the microfibrils. Because this representation was assumed to apply to both the core-shell and interconnected-interface models of compatibilization, the constrained-yielding approach was used without specific reference to the microstructure of the interface. In extending the discussion to fracture, however, it is useful to consider the interfacial-deformation mechanisms. Tensile deformation culminated in catastrophic fracture when the microfibrillated interface failed. This was inferred from the quasi-brittle fracture behavior of the uncompatibilized blend with VPS of 0.5, which indicated that the reduced load-bearing cross section after interfacial debonding could not support plastic deformation. Accordingly, the ultimate properties of the compatibilized blend depended on interfacial char-... 

The behavior of this circuit differs qualitatively from the previous one, because conductances gi, g2 and capacitances Ci, C2 are in parallel. Thus the circuit is equivalent to that of Figure 4.1.3b, which shows only one relaxation. For the microstructure of Figure 4.12b the individual relaxations cannot be resolved by any method, graphical, CNLS, or other. Although at first glance this model would appear to be as plausible as the series layer model, it fails to describe the behavior of grain boundaries in ceramics. 

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