Stiffness parameters

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

Numerical calculations inspired in the ZK method for stars have also been applied for the description of the dynamics of model dendrimers. La Ferla [232] used a freely-rotating model, including a topology-dependence stiffness parameter and preaveraged HI. With this model, he obtained a complex analytical expression for the mean size. Cai and Chen [233] used a Gaussian model without HI and performed a detailed analysis of the relaxation motions. They investigated the diffusion of the center of mass, the relaxation of the center of mass position relative to the core monomer, and also the rotational and internal modes. 

The group-theoretical stiffness parameters can be expressed in the conventional (cubic) elastic stiffness coefficients ... 

In performing such experiments on isotropic materials, one is accustomed to express the elastic stiffness parameters in the experimentally more readily accessible technical parameters E (Young s modulus) and v (Poisson ratio). The relative change in length, in the direction of the tensile stress a is, by definition, given by (Al/t)i — a/E, whereas v = (Af/ )x/( A / )u. For several magnetostrictive films and substrates, E and v values are listed in table 1. Some useful relations are ... 

Earlier69 we obtained the following empirical equation relating Tg to the chain stiffness parameter ... 

Using Eqs. (25)-(28), one obtains for the lattice spacing Z, the chain stiffness parameter to and the number of segments y the expressions... 

According to Eq. (33), the chain stiffness parameter to decreases weakly with the chain length. Consequently, the polymer chain becomes slightly stiffer with increasing chain length, which is unreasonable. For this reason, an average of the values provided by Eq. (33) for the four PI chains employed was used. For the PI chains, Ms= 68 g/mol, lb = 5.07 A, p = 0.913 g/cm3 45 consequently Z=4.94 A and w=0.29 (calculated as the arithmetic average). All the parameters for PS and PI chains are listed in Table I, which also includes three polymer-solvent interaction parameters for reasons explained below. 

The stiffness parameter C55 has, in effect, been increased by the factor (1 + K ) — an effect known as piezoelectric stiffening. The factor is the electromechanical coupling coefficient for the jt-propagating, z-polarized plane wave ... 

The loss factor peak which occurs at Yopt can be located in frequency by the appropriate choice of the parameters that define y (see below). The maximum loss factor Tjmax the peak is governed by T 2 the loss factor of the viscoelastic layer and by a "stiffness" parameter Y simply defined as (17)... 

The parameter Y depends only on the properties of the base and constraining layers. Although we have called it a stiffness parameter, Y does become dependent only on geometry when the two outside layers have equal elastic moduli. 

Figure 7. Maximum Loss Factor versus Stiffness Parameter for Constrained Viscoelastic Layer Treatments (Adapted from ref. 10)...

In the equation above, bend is the stiffness parameter, while and 0 stand for the instantaneous and the average angles between consecutive bonds, respectively. Apart from the two conservative interactions, DPD particles experience pairwise dissipative forces, depending on the relative speed of the particles, and pairwise random forces that depend on their relative distances. The properties of the dissipative and the stochastic forces are chosen to ensure local momentum conservation and that the equilibrium states of the system comply with the Boltzmann distribution of the ensemble [96],... 

Spin-wave stiffness parameters D for Pt3Cr and Pt3MnCr (Williams et al. 1981) and Pt3Mn (Paul and Stirling 1979). 

All of these stress relations are derived from Eq. (1.8). They are valid therefore only for moderate deformations of the network that is, for deformations sufficiently small for the chain tensions to be linearly related to their end-to-end distances r (Eq. (1.1)). Unfortunately, no correspondingly simple expression can be formulated for W using Eq. (1.5), the relationship for large strains of the constiment chains, in which the molecular stiffness parameter reappears. Instead, a variety of series approximations must be used, as in Eq. (1.6), to give close approximations to the behavior of rubber networks under large strains (Arruda and Boyce, 1993). 

Equations (6.9) or (6.4) and (6.10) provide the stiffnesses of a ply with hbers oriented at an angle 6 in the laminate coordinate system.The above discussion concentrated on plane stress conditions. In a three-dimensional situation, a 0° ply is dehned with nine stiffness parameters En, E22, E33, G12, G13, G23, J i2, 13, and i 23. Here, as before, 1 denotes the direction parallel to the hbers, 2 is perpendicular to the hbers, and 3 is the out-of-plane direction. Eor orthotropic plies, E22 = E33,... 

There are two main purposes of model updating or system identification. One common goal is to identify physical parameters, e.g., stiffness of a structural element or the diffusion rate of an air pollutant. These identified parameters can be further used as indicator for the status of the system or phenomenon. For example, the stiffness parameter of a structural member can be monitored from time to time and an abnormal reduction indicates possible damage of the member. However, reduction may be simply due to statistical uncertainty. Therefore, it is necessary to quantify the uncertainty of the estimation so that one can distinguish whether the parameter change is due to deterioration of the structural member. In this case, it is desirable to obtain a narrow distribution of the parameter so that small changes can be detected with a high level of confidence. 

The prior PDF p(0 C) is taken to be independent log-normal PDFs with means of 0.9 and 1.2 and unit variance. Using the modal data V, the updated PDF for the stiffness parameter vector 0 is formulated as ... 

If it is assumed that only one sensor (at either the first or second floor) was used during the modal testing, only the modal frequencies can be identified. In this case, the two stiffness parameters are locally identifiable and the normalized modal goodness-of-fit function is given... 

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