The stiffness problem

In this method, instead of working with 4> one works with a new set of variables that are combinations of the fundamental solutions 4> and 3. These new variables all vary with y at the identical rate, removing the stiffness problem. For the Orr-Sommerfeld equation the new variables are (for details about these new variables see Drazin Reid (1981), Sengupta (1992)),... 

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

For the stiff problems, based on multivalue algorithms of the Gear family ... 

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 the model problem described earlier, replace the stiff spring potential Vs = K r - 1)2/2 by... 

To be more precise, this error occurs in the limit /c — oo with Ef = 0(1) and step-size k such that k /ii = const. 3> 1. This error does not occur if Ef = 0 for the analytic problem, i.e., in case there is no vibrational energy in the stiff spring which implies V,. = U. 

Frontal solution requires very intricate bookkeeping for tracking coefficients and making sure that all of the stiffness equations have been assembled and fully reduced. The process time requirement in frontal solvers is hence larger than a straightforward band solver for equal size problems. 

Some cast (unoriented) polypropylene film is produced. Its clarity and heat sealabiUty make it ideal for textile packaging and overwrap. The use of copolymers with ethylene improves low temperature impact, which is the primary problem with unoriented PP film. Orientation improves the clarity and stiffness of polypropylene film, and dramatically increases low temperature impact strength. BOPP film, however, is not readily heat-sealed and so is coextmded or coated with resins with lower melting points than the polypropylene shrinkage temperature. These layers may also provide improved barrier properties. 

S. A. Ambartsumyan, The Axisymmetric Problem of a Circular Cylindrical Shell Made of Material with Different Stiffness in Tension etnd Compression, Izvesttya Akademii Nauk SSSR Mekhanika, No. 4, 1965, pp. 77-85 English translation N69-11070, STAR. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

The objective of this chapter is to address introductory sketches of some fundamental behavior issues that affect the performance of composite materials and structures. The basic questions are, given the mechanics of the problem (primarily the state of stress) and the materials basis of the problem (essentially the state of the material) (1) what are the stiffnesses, (2) what are the strengths, and (3) what is the life of the composite material or structure as influenced by the behavioral or environmental issues in Figure 6-1 ... 

A simplified performance index for stiffness is readily obtained from the essentials of micromechanics theory (see, for example. Chapter 3). The fundamental engineering constants for a unidirectionally reinforced lamina, ., 2, v.,2, and G.,2, are easily analyzed with simple back-of-the-envelope calculations that reveal which engineering constants are dominated by the fiber properties, which by the matrix properties, and which are not dominated by either fiber or matrix properties. Recall that the fiber-direction modulus, is fiber-dominated. Moreover, both the modulus transverse to the fibers, 2, and the shear modulus, G12. are matrix-dominated. Finally, the Poisson s ratio, v.,2, is neither fiber-dominated nor matrix-dominated. Accordingly, if for design purposes the matrix has been selected but the value of 1 is insufficient, then another more-capable fiber system is necessary. Flowever, if 2 and/or G12 are insufficient, then selection of a different fiber system will do no practical good. The actual problem is the matrix systemi The same arguments apply to variations in the relative percentages of fiber and matrix for a fixed material system. 

The first problem area of the so-called anisotropic analysis will be broken down into two subareas shear-extension coupling and bend-twist coupling. We have already observed for the most complicated laminate in the design philosophy proposed earlier that the A,q and A26 stiffnesses are both zero. There is no shear-extension coupling in the context of that philosophy. However, in contemporary composite structures analyses, it is relatively easy to include the treatment of shear-extension coupling, so you should not be overwhelmed by that behavioral aspect or by the calculation of its influence. 

The next problem area is transverse shearing effects. There are some distinct characteristics of composite materials that bear very strongly on this situation because for a composite material the transverse shearing stiffness, i.e., perpendicular to the plane of the fibers, is considerably less than the shear stiffness in the plane of the fibers. There is a shear stiffness for a composite material in a plane that involves one fiber direction. Shear involves two directions always, and one of the directions in the plane is a fiber direction. That shear stiffness is quite a bit bigger than the shear stiffness in a plane which is perpendicular to the axis of the fibers. The shear stiffness in a plane which is perpendicular to the axis of the fibers is matrix-dominated and hardly fiber-influenced. Therefore, that shear stiffness is much closer to that of the matrix material itself (a low value compared to the in-plane shear stiffness). 

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

We have used various integrators (e.g., Runga-Kutta, velocity verlet, midpoint) to propagate the coupled set of first-order differential equations Eqs. (2.8) and (2.9) for the parameters of the Gaussian basis functions and Eq. (2.11) for the complex amplitudes. The specific choice is guided by the complexity of the problem and/or the stiffness of the differential equations. 

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