Stiffnesses restrictions

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

Irrespective of the micromechanical stiffness approach used, the basic restrictions on the composite material that can be treated are ... 

A collection of the basic building block, a lamina, was bonded together to form a laminate in Chapter 4. The behavior restrictions were covered in the section on classical lamination theory. Special cases of laminates were discussed to learn about laminate characteristics and behavior. Predicted and measured laminate stiffnesses were favorably compared to give credence to classical lamination theory. Then, the strength of laminates was discussed and found to be reasonably predictable. Finally, interlaminar stresses were analyzed because of their apparent strong influence on laminate strength (and life). 

MW fraction increases the melt flow, thus improving the processability but at the cost of toughness, stiffness, and stress crack resistance. In addition, the improvement in performance through narrowing the MWD is restricted by the catalyst, the process hardware, and the process control limitations. Dow has developed a reactor grade HDPE of optimized breadth, peak, and shape of MWD... 

A higher TYPE 2 restriction caused by a better mixing in the system with PTM 0(650) would increase the stiffness of the oligomeric chain, and this should result in a higher modulus. As the PTMO molecular weight increases, the encapsulation of oligomeric chains by condensed TEOS species would become more difficult. Such a decrease in the TYPE 2 restrictions would make the PTMO chains less stiff and, as a result, cause the modulus to decrease. 

Causes of diastolic dysfunction (restriction in ventricular filling) are increased ventricular stiffness, ventricular hypertrophy, infiltrative myocardial diseases, myocardial ischemia and infarction, mitral or tricuspid valve stenosis, and pericardial disease (e.g., pericarditis, pericardial tamponade). 

Restrictions which may exist for the choice of a commercial reactor need not be imposed at the development stage. In some cases, a reactor of one type may be best for acquiring data in model characterisation, whereas a reactor of another type might be more suitable for full-scale production. (The cautions expressed in Sect. 4 must be taken into account.) Continuous flow back-mixed reactors can be very useful for kinetic studies because the absence of concentration gradients can reduce uncertainties in concentration measurements. When these reactors have attained a steady state, many of the problems associated with stiffness (see above) can be avoided. 

Finally, the use of low-molecular-weight species to improve flow properties called plasticizers normally reduces stiffness, hardness, and brittleness. Plasticization is usually restricted to amorphous polymers or polymers with a low degree of crystallinity because of the limited compatibility of plasticizers with highly crystalline polymers. Other additives, such as antioxidants, do not affect the mechanical properties significantly by themselves, but can substantially improve property retention over long periods of time, particularly where the polymer is subject to environmental degradation. 

The Tg is related to chain stiffness and the geometry of the polymer chain. Flexible polymers with methylene and oxygen atoms in the chain, such as polyethylene, polyoxymethylene, and polysiloxane (silicone), have relatively low Tg values. The Tg of polyoxymethylene is somewhat higher than would be anticipated because of the dipole character of the C—O—C group, which increases the intermolecular forces and restricts segmental motion. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

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