Material parameter representative examples

A general method of applying viscoelasticity theory to unstable (changing) materials in varying temperature fields was proposed in a number of publications (see, for example Ref.135). In this approach, the state of a material is represented by the factor jr, which is a function of a set of "structural" parameters... 

If n is restricted to the y-z plane during the shear, a steady windup solution obtains for low shear rates, with n rotating the most at the center and less toward the walls. This windup picture becomes unstable at a critical shear rate, where the director tumbles discon-tinuously to a new solution with reduced elastic energy. The critical Er depends on material parameters, and falls roughly between 10 and 100. Mathematically, this instability is represented by the existence of multiple in-plane solutions at certain ranges of the shear rate. An example is shown in Fig. 2. 

Note that in the context of both the Nabarro-Herring analysis and that of Coble, the strain rate is found to vary linearly with the applied stress. In particular, this result is consistent with dubbing this process diffusional viscosity as did Herring (1950) in his original work. Indeed, the identification of a material parameter such as the viscosity and its associated scaling properties with grain size and temperature represents another example of the type of micro-macro connection... 

Inclusion of the HF exchange, combined with the optimisation of the parameters included in the hybrid functionals, yielded a noticeable improvement of DFT results it has in this way been crucial in achieving a working formulation of DFT that is accurate enough to satisfy the quantum chemistry needs for many applications. Combined with its relatively low computational cost, this development has transformed DFT into the method of choice for the ab-initio calculation of materials properties, especially for large systems of practical interest. Several critical cases remain, however, where standard one-electron Hamiltonians fail to reach sufficient accuracy to interpret and/or direct experiments. Below we shall consider some representative examples in the solid state. 

The relaxation period defines the behavior of the system, in accordance with the Maxwell model with respect to the timescale of the applied stress. If the time t during which stress is applied is greater than the relaxation period, that is, t > t the system has properties similar to those of a viscous liquid, while at t t the system behaves like an elastic solid. The flow of glaciers and other processes of strain development in mountains and cliffs are representative examples of such behavior. In rheology, the ratio of a material s characteristic relaxation time to the characteristic flow time is referred to as the Deborah number. This parameter plays an important role in describing the response of various materials to different stresses. 

Again each term on the right hand side of Elq. 2.40 represents a double summation and each coefficient of strain is an independent set of material parameters. Thus, many more than 81 parameters may be required to represent a nonlinear heterogeneous and anisotropic material. Further, for viscoelastic materials, these material parameters are time dependent. The introduction of the assumption of linearity reduces the number of parameters to 81 while homogeneity removes their spatial variation (i.e., the parameters are now constants). Symmetry of the stress and strain tensors (matrices) reduces the number of constants to 36. The existence of a strain energy potential reduces the number of constants to 21. Material symmetry reduces the number of constants further. For example, an orthotropic material, one with three planes of material symmetry, has only 9 constants and an isotropic material, one with a center of symmetry, has only two independent constants (and Eq. 2.39 reduces to Eq. 2.28). Now it is easy to see why the assumptions of linearity, homogeneity and isotropy are used for most engineering analyses. 

In the study, a gene represents a real number (e.g., the reorder point s, or the order-up-to-point S, if an (s, S) policy is to be optimised). A chromosome represents a complete set of all control parameters. For example, if (s, S) policies are applied for raw material procurement and finished goods production, a chromosome will consist of eight real numbers since there are four types of raw materials and one type of finished goods. The real parameter GA can use the objective fitness value directly. The SOGA program is applied to optimise one objective, for example, SC total cost or GSL. 

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