Plate constitutive equations

For the situation where the loading is applied off the fibre axis, then the above approach involving the Plate Constitutive Equations can be used but it is necessary to use the transformed stiffness matrix terms Q. 

The use of the Plate Constitutive Equations is illustrated in the following Examples. 

As earlier we may group equations (3.36) and (3.39) to give the Plate Constitutive Equation as... 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

The solution method using the Plate Constitutive Equation is therefore straightforward and very powerful. Generally a computer is needed to handle... 

The Plate Constitutive equations can be used for curved plates provided the radius of curvature is large relative to the thickness (typically r/h > 50). They can also be used to analyse laminates made up of materials other than unidirectional fibres, eg layers which are isotropic or made from woven fabrics can be analysed by inserting the relevant properties for the local 1-2 directions. Sandwich panels can also be analysed by using a thickness and appropriate properties for the core material. These types of situation are considered in the following Examples. 

Using the constitutive equation, we can also compute the force it takes to move the upper plate (barrel)... 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

With viscoelastic models used by an increasing number of researchers, time and temperature dependence, as well as strain hardening and nonisotropic properties of the deformed parison can, in principle, be accounted for. Kouba and Vlachopoulos (97) used the K-BKZ viscoelastic constitutive equation to model both thermoforming and parison membrane stretching using two-dimensional plate elements in three-dimensional space. Debbaut et al. (98,99) performed nonisothermal simulations using the Giesekus constitutive equation. 

For example, when we consider the design of specialty chemical, polymer, biological, electronic materials, etc. processes, the separation units are usually described by transport-limited models, rather than the thermodynamically limited models encountered in petrochemical processes (flash drums, plate distillations, plate absorbers, extractions, etc.). Thus, from a design perspective, we need to estimate vapor-liquid-solid equilibria, as well as transport coefficients. Similarly, we need to estimate reaction kinetic models for all kinds of reactors, for example, chemical, polymer, biological, and electronic materials reactors, as well as crystallization kinetics, based on the molecular structures of the components present. Furthermore, it will be necessary to estimate constitutive equations for the complex materials we will encounter in new processes. 

Since steel backup and hot copper plates are assumed to exhibit thermo-elastic and -elastic-plastic respectively, isotropic linear elastic stress-strain relation can be solved by constitutive equation (1) using finite element method. 

Equation 2.3.22 is independent of the constitutive equation because of the small angle. The result is a homogeneous shear field, like simple shear between sliding parallel plates or between closely fitting cylinders. Because stress and deformation rates can be determined independent of a constitutive equation, these flows are very useful as rheometers and are discussed further in Chapter S. 

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