Anisotropic analysis

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. 

Bend-twist coupling is a totally different animal. The governing stiffnesses, D g and D2g, simply are never zero for any laminate that is more complicated than a cross-ply laminate. You cannot force those stiffnesses to go to zero unless you do something else to the laminate. You can make them go to zero if you let the laminate be unsymmetric, but that is robbing Peter to pay Paul. In fact, it is not very difficult in most contemporary analyses to include the influence of those bend-twist cou- 

The effect of the specific values of the B j can be readily calculated for some simple laminates and can be calculated without significant difficulty for many more complex laminates. The influence of bending-extension coupling can be evaluated by use of the reduced bending stiffness approximation suggested by Ashton [7-20]. If you examine the matrix manipulations for the inversion of the force-strain-curvature and moment-strain-curvature relations (see Section 4.4), you will find a definition that relates to the reduced bending stiffness approximation. You will find that you could use as the bending stiffness of the entire structure, 

The next problem area of micromechanics is initially very attractive in some respects. We look to the fundamental definition of a composite material made up in this case of, say, a fiber and a matrix and attempt to actually design that material. Let us change the proportions of fibers and matrix so that we get the kind of material behavior characteristics we want. That objective is admirable, but achieving that objective in all cases is not entirely realistic. 

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Regarding the macromechanics perspective, assuming the material to be almost uniform, its anisotropy can be exploited as an advantage. The average behavior of the material can be predicted and controlled by knowing the properties of its constituents. Nevertheless, the anisotropic analysis is more complex and more dependent on the measurement procedures, whereas the analysis of conventional materials is easier due to their isotropy and uniformity. 

In addition anisotropic scaling allows for three possibilities for the fixed point of the nematic splay elastic constant Xi = 0, implying = vy/vj = 1, Xj finite, implying A = 2, and Xj infinite, implying A > 2. The Nelson-Toner anisotropic analysis [19] predicts A = 2. All options imply either a strong anisotropy (A = > 2 or no anisotropy at all A = = 1. Experimentally, a weak anisotropy... 

LeRoy R J and van Kranendonk J 1974 Anisotropic intermolecular potentials from an analysis of spectra of H2- and D2-inert gas complexes J. Chem. Phys. 61 4750... 

Other interference-produced colors falling into this section include doubly refracting materials such as anisotropic crystals and strained isotropic media between polarizers, as in photoelastic stress analysis and in the petrological microscope. 

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

Conventionally RAIRS has been used for both qualitative and quantitative characterization of adsorbed molecules or films on mirror-like (metallic) substrates [4.265]. In the last decade the applicability of RAIRS to the quantitative analysis of adsorbates on non-metallic surfaces (e.g. semiconductors, glasses [4.267], and water [4.273]) has also been proven. The classical three-phase model for a thin isotropic adsorbate layer on a metallic surface was developed by Greenler [4.265, 4.272]. Calculations for the model have been extended to include description of anisotropic layers on dielectric substrates [4.274-4.276]. 

In order to describe completely the state of triaxial (as opposed to biaxial) stress in an anisotropic material, the compliance matrix will have 36 terms. The reader is referred to the more advanced composites texts listed in the Bibliography if these more complex states of stress are of interest. It is conventional to be consistent and use the terminology of the more general analysis even when one is considering the simpler plane stress situation. Hence, the compliance matrix [5] has the terms... 

The previous analysis has shown that the properties of unidirectional fibre composites are highly anisotropic. To alleviate this problem, it is common to build up laminates consisting of stacks of unidirectional lamina arranged at different orientations. Clearly many permutations are possible in terms of the numbers of layers (or plies) and the relative orientation of the fibres in each... 

The pressure is to be identified as the component of stress in the direction of wave propagation if the stress tensor is anisotropic (nonhydrostatic). Through application of Eqs. (2.1) for various experiments, high pressure stress-volume states are directly determined, and, with assumptions on thermal properties and temperature, equations of state can be determined from data analysis. As shown in Fig. 2.3, determination of individual stress-volume states for shock-compressed solids results in a set of single end state points characterized by a line connecting the shock state to the unshocked state. Thus, the observed stress-volume points, the Hugoniot, determined do not represent a stress-volume path for a continuous loading. 

The special single-layered configurations treated in this section are isotropic, specially orthotropic, generally orthotropic, and anisotropic. The generally orthotropic configuration cannot, of course, be distinguished from an anisotropic layer from the analysis point of view, but does have only the four independent material properties of an orthotropic material. 

James M. Whitney, Structural Analysis of Laminated Anisotropic Piates, Technomic, Lancaster, Pennsylvania, 1987. See also J. E. Ashton and J. M. Whitney, Theory of Laminated Plates, Technomic, Westport, Connecticut, 1970. 

J. E. Ashton, Anisotropic Plate Analysis, General Dynamics Research and Engineering Report, FZM-4899, 12 October 1967. 

J. M. Whitney and A. W. Leissa, Analysis of Heterogeneous Anisotropic Plates, Journal of Applied Mechanics, June 1969, pp. 261-266. 

J. E. Ashton and M. E. Waddoups, Analysis of Anisotropic Plates, Journal of Composite Materials, January 1969, pp. 148-165. 

J. M. Whitney, On the Analysis of Anisotropic Rectangular Plates, Air Force Materials Laboratory Technical Report AFML-TR-72-76, August 1972. 

The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

Wave propagation in an inhomogeneous anisotropic material such as a fiber-reinforced composite material is a very complex subject. However, its study is motivated by many important applications such as the use of fiber-reinforced composites in reentry vehicle nosetips, heatshields, and other protective systems. Chou [6-56] gives an introduction to analysis of wave propagation in composite materials. Others have applied wave propagation theory to shell stress problems. 

The study of composite materials actually involves many topics, such as, for example, manufacturing processes, anisotropic elasticity, strength of anisotropic materials, and micromechanics. Truly, no one individual can claim a complete understanding of all these areas. Any practitioner will be likely to limit his attention to one or two subareas of the broad possibilities of analysis versus design, micromechanics versus macromechanics, etc. 

Definitive proof of the structure of porphine in the solid state awaits a variable-temperature crystallographic (X-ray or neutron diffraction) study the analysis of the anisotropic displacement factors (ADP) should disclose any rotational motion or its absence as well as determine the positions of the inner hydrogens. A search in the September 1998 version of the Cambridge Structural Database [CSD (91MI187)] showed that the only structures of porphine (codename PORPIN) were obtained in 1965 and 1972. 

The last result was obtained independently in [27,269], In the logarithmic scale of Fig. 6.3 the dependence (6.25) is linear in both cases, but its slope in the isotropic case is opposite to that in the anisotropic case. This difference makes it possible to perform self-consistent verification of the theories. Unfortunately, independent information on xj is rather rare. It can be obtained from NMR investigations, or from analysis of the wings of the spectrum (6.20). Since both tasks are rather complex,... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

It is now well-established that for atomic fluids, far from the critical point, the atomic organisation is dictated by the repulsive forces while the longer range attractive forces serve to maintain the high density [34]. The investigation of systems of hard spheres can therefore be used as simple models for atomic systems they also serve as a basis for a thermodynamic perturbation analysis to introduce the attractive forces in a van der Waals-like approach [35]. In consequence it is to be expected that the anisotropic repulsive forces would be responsible for the structure of liquid crystal phases and numerous simulation studies of hard objects have been undertaken to explore this possibility [36]. 

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