Principal material directions determination

What has been accomplished in preceding sections on stiffness relationships serves as the basis for determination of the actual stress field what remains is the definition of the allowable stress field. The first step in such a definition is the establishment of allowable stresses or strengths in the principal material directions. Such information is basic to the study of strength of an orthotropic lamina. 

Determination of the fourth-rank tensor term F. 2 remains. Basically, F.,2 cannot be found from any uniaxial test in the principal material directions. Instead, a biaxial test must be used. This fact should not be surprising because F-,2 is the coefficient of the product of a. and 02 in the failure criterion. Equation (2.140). Thus, for example, we can impose a state of biaxial tension described by a, = C2 = c and all other stresses are zero. Accordingly, from Equation (2.140),... 

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 topic of invariant transformed reduced stiffnesses of orthotropic and anisotropic laminae was introduced in Section 2.7. There, the rearrangement of stiffness transformation equations by Tsai and Pagano [7-16 and 7-17] was shown to be quite advantageous. In particular, certain invariant components of the lamina stiffnesses become apparent and are heipful in determining how the iamina stiffnesses change with transformation to non-principal material directions that are essential for a laminate. 

The magnitude of the elastic moduli obtained for an anisotropic material will depend on the orientation of the coordinates used to describe the material elastic response. However, if the material elastic moduli are known for coordinates aligned with the principal material directions, then the elastic moduli for any other orientation can be determined through appropriate transformation equations. Thus, only four elastic constants are needed in order to fully characterize the in-plane maaoscopic elastic response of an orthotropic lamina. The reference coordinates in the plane of the lamina are aligned with longitudinal axis (L) parallel to the fibers, and the transverse axis (7) perpendicnlar to the fibers. The engineering orthotropic elastic moduli of the lamina defined earher are... 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

In Section 8.3, the lamina macrostructural elastic moduli have been defined for the principal material directions. The elastic properties are determined experimentally from uniaxial stress states. If these experiments are carried through to failure of the test specimen, then one also obtains the following macromechanical failure strength properties of the lamina defined in principal material coordinates ... 

In order to apply the maximum stress theory of failure, the stress components must be determined in principal material directions. Using the transformation Equation 8.56, the following relationships are obtained ... 

X-Ray diffraction (XRD) is the principal means of determining the structures of crystals. It is a technique in which a collimated X-ray beam is directed at a single crystal of the material under investigation or, as is more usual in the study of fat crystals, a sample comprising a large number of randomly orientated crystals. The latter variant, which is the most commonly used generally, is called powder diffraction (Cullity, 1956). Both variants yield essentially the same basic information about structure after data analysis. The following descriptions relate to powder diffraction specifically. 

The Tresca yield criterion or maocimum shear stress criterion is not directly based on the considerations of the previous sections, but it fulfils them nevertheless. It states that the maximum shear stress in the material point determines yielding. This maximum shear stress can be determined graphically using Mohr s circle, see figure 2.3 on page 34. The maximum principal stress is denoted as a, the intermediate value as au, and the smallest as crni- The maximum shear stress is... 

Low frequency dielectric studies on smectic C, F and I phases are complicated by the intrinsic biaxiality of these phases. It is possible to use dielectric measurements to determine the tilt angle in SmC materials [27], but of more interest is the direct determination of the three principal components of the dielectric tensor, since such measurements can give additional information on the local molecular organization from Eq. (29). The orientation of the principal axes for tilted smectic phases is not determined by symmetry, except that one principal axis coincides with the C2 rotation axis perpendicular to the tilt-plane. It is assumed that a further principal axis lies along the tilt direction, and this appears to be justified by experiment the orientation of the axes are indicated in Fig. 9. 

The results regarding socioeconomic gradients undermines the hypothesis that the principal social class influence on health is material deprivation. In fact, the social class gradient in health cuts deeply into the affluent middle classes. The implication is that the conditions under which people live can affect human health directly, and not only through material deprivation. Early childhood experience, one s place in the social environment, and the experiences of daily life must be powerful determinants of the length and healthfulness of life (Kelly et al., 1997, p. 438). 

The optical properties of the components of petroleum have been of major importance in connection with their identification and in the determination of purity. The primary effort has been directed to the study of pure hydrocarbons and only limited work has been concerned with the prediction of the index of refraction and the specific rotation of hydrocarbon mixtures. Table V summarizes the optical properties of a number of the principal components of petroleum. Only a few references to the optical properties of pure hydrocarbons of primary interest to the analyst have been included. Developments (9) in refractometers have materially increased the potentialities of the index of refraction measurements at atmospheric pressure as an analytical method. Consideration of the pertinent data in this field is beyond the scope of the present discussion. Reviews of developments in infrared (24, 26) and mass spectrometry (68) are available. 

The principal strain rates are eigenvalues of the strain-rate tensor (matrix). As described more fully in Section A.21, the direction cosines that describe the orientation of the principal strain rates are the eigenvectors associated with the eigenvalues. In solving practical fluids problems, there is rarely a need to determine the principal strain rates or their orientations. Rather, these notions are used theoretically with the Stokes postulates to form general relationships between the strain-rate and stress tensors. It is perhaps worth noting that in solid mechanics, the principal stresses and strains have practical utility in understanding the behavior of materials and structures. 

The saturation magnetization Ms is a specific constant for the material and for magnetic iron oxides is principally determined by the Fe2 + ion content. The ratio of remanent magnetization to saturation magnetization (Mr/Ms) for the tape depends mainly on the orientation of the pigment needles with respect to the longitudinal direction of the tape, and should approach the theoretical maximum value of unity as closely as possible. 

A delta function, 5(f),is a distribution that equals zero everywhere except where its argument is zero, where it has an infinite singularity. It has the property f f r)5(r- ro)dr = /(ff0) so it also follows that /S(r — ro)dr = 1. The singularity of 5(f — fo) is located at F o-BThis technique can be used to measure the diffusivity in anisotropic materials, as described in Section 4.5. Measurements of the concentration profile in the principal directions can be used to determine the entire diffusion tensor. 

Raman spectroscopy can also directly benefit TE analysis by non-invasively monitoring the growth and development of ECM by different cells on a multitude of scaffold materials exposed to various stimuli (e.g. growth factors, mechanical forces and/or oxygen pressures). Indeed the non-invasive nature of Raman spectroscopy enables the determination of the rate of ECM formation and the biochemical constituents of the ECM formed. Univariate (peak area, peak ratios, etc.) and multivariate analytical techniques (e.g. principal component analysis (PCA)) can be used to determine if there are any significant differences between the ECM formed on various scaffolds and/or cultured with different environmental parameters, and what these biochemical differences are. Least square (LS) modelling, for example, could allow the quantification of the relative components of the ECM formed (Fig. 18.3) [4, 38],... 

Figure 1.13 shows the spectrum of a hexapeptide [181] as its permethy-lated derivative with the ions indicating the amino acid order. The rapid decrease in the intensity of the important sequence-determining ions at higher mass is evident and a principal reason why MS often requires several times as much material as the common micro wet chemical methods for sequencing small peptides. The chemical ionization mass spectra of some N -acyl permethylated simple peptides show a much more even distribution of the sequencing peaks and hence require a lower sample level than for El spectra [187]. This may well prove of value in the future. Cl has also been applied directly to peptides [188] where up to six amino acids units have been introduced by the direct insertion probe. 

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