Physics based constitutive model

Physically based constitutive models that accurately capture the behavior of fluoropolymers. 

With increased understanding of various types of SMPs, physics based constitutive modeling is possible. With such models, fewer parameters will be needed and the predictability of the models will be more reliable and accurate. Also, a physics based model can be easily integrated into finite element modeling (FEM) so that SMP based engineering structures and devices can be analyzed and designed. This will be a direction for future studies. 

Development of Physics Based Constitutive Modeling of Shape Memory Polymers... 

A particular topic approached in this book is new insight into the physical origin of inelastic effects in reinforced elastomers, to assist with the development of physically-based constitutive models. 

In general, SMPF is perceived as a two-phase composite material with a crystalline phase mixed with an amorphous phase. A multiscale viscoplasticity theory is developed. The amorphous phase is modeled using the Boyce model, while the crystalline phase is modeled using the Hutchinson model. Under an isostrain assumption, the micromechanics approach is used to assemble the microscale RVE. The kinematic relation is used to link the micro-mechanics constitutive relation to the macroscopic constitutive law. The proposed theory takes into account the stress induced crystallization process and the initial morphological texture, while the polymeric texture is updated based on the apphed stresses. The related computational issue is discussed. The predictabihty of the model is vahdated by comparison wifli test results. It is expected that more accurate measurement of the stress and strain in the SMPF with large deformation may further enhance the predictability of the developed model. It is also desired to reduce the number of material parameters in the model. In other words, a deeper understanding and physics based theoretical modeling are needed. 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

Modeling is also a requirement for the design space. However, what constitutes a model can vaiy from an almost totally empirical model to a first principles model All may be valid if the assumptions upon which the model was created are clear and adhered to. For example, the model presented above is an empirical model based upon selection of variables that seem logical based upon the science and statistical analysis of the data collected. If we had a physics equation (constitutive relationship) and the ability to predict all of the variables, it would be a first principles model. In between empirical and first principles are so called hybrid" models that may have known relationtihips between variables but require calibration or determination of coefficients. The differ-ences are that ... 

Heinrich, G., Kaliske, M., 1997. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. CompuL Theor. Polym. Sci. 7 (3-4), 227-241. Heinrich, G., Straube, E., Helmis, G., 1988. Rubber Elasticity of Polymer Networks Theories Polymer Physics. Springer, Berlin/Heidelberg, pp. 33-87. 

As it can be inferred from this description, the solution of Eq. 49 leads to additional computations to help ensure that the stress-strain relationship is maintained according to the nonlinear constitutive model of choice. This requires the use of implicit solution schemes, as opposed to the explicit approach used in the elastic and viscoelastic problems. This additional effort explains why considering the material s plastic behavior in physics-based ground-motion simulations in 3D has often been ignored or simplified using hybrid and indirect approaches. Future developments, however, are likely to revert this... 

The fundamental concept of the material clock or reduced time is similar to the principle described above in the discussion of time-temperature superposition. In the mechanical constitutive models, however, the change in the stress or deformation induces a shift in the material relaxation time. The fact that the time depends on the state of stress (or strain) or on its history leads to additional non-linearities in behavior from what is expected with, eg, the K-BKZ model. Physical explanations for the shifting material time are often based on free-volume ideas that are often invoked to explain time-temperature superposition. In addition, entropy changes have been invoked as have stress-activated processes. 

The main tools used to provide global projections of future climate are general circulation models (GCMs). These are mathematical models based on fundamental physical laws and thus constitute dynamical representations of the climate system. Computational constraints impose a limitation on the resolution that it is possible to realise with such models, and so some unresolved processes are parameterised within the models. This includes many key processes that control climate sensitivity such as clouds, vegetation and oceanic convection [19] of which scientific understanding is still incomplete. 

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

In contrast to Section IV,M, where the turbulent diffusivity was employed to derive an expression for the mass transfer coefficient, in this section expression (401), which is based on a physical model, constitutes the starting point. Concerning the renewal frequency s, the following dimensional considerations can lead to useful expressions. The state of turbulence near the interface can be characterized by a characteristic velocity ua = (gSf)i/2, the dynamic viscosity rj, the surface tension a, and the density p. Therefore... 

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