Orthotropic

Most of the laminates used for rigid printed circuit boards have been classified, by the National Electrical Manufacturers Association (NEMA), according to the combination of properties that determine the suitabiHty of a laminate for a particular use. Eiber reinforcements make laminate-effective properties orthotropic. 

Because the fibers generally are anisotropic, they tend to be deposited on the wire in layers under shear. There is Htde tendency for fibers to be oriented in an out-of-plane direction, except for small undulations where one fiber crosses or passes beneath another. The layered stmcture results in the different properties measured in the thickness direction as compared to those measured in the in-plane direction. The orthotropic behavior of paper is observed in most paper properties and especially in the electrical and mechanical properties. 

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

In contrast, composite materials are often both inhomogeneous (or nonhomogeneous or heterogeneous — the three terms can be used interchangeably) and nonisotropic (orthotropic or, more generally, anisotropic, but the words are not interchangeable) ... 

An orthotropic body has material properties that are different in three mutually perpendicular directions at a point in the body and, further, has three mutually perpendicular planes of material property symmetry. Thus, the properties depend on orientation at a point in the body. 

The inherent anisotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different from those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loadings of normal stress and shear stress is shown in Figure 1-4 and discussed in the following paragraphs. 

Most simple materia characterization tests are perfomned with a known load or stress. The resulting displacement or strain is then measured. The engineering constants are generally the slope of a stress-strain curve (e.g., E = o/e) or the slope of a strain-strain curve (e.g., v = -ey/ej5 for Ox = a and all other stresses are zero). Thus, the components of the compliance (Sy) matrix are determined more directly than those of the stiffness (Cy) matrix. For an orthotropic material, the compliance matrix components in terms of the engineering constants are... 

Note that an orthotropic material that is stressed in principal material coordinates (the 1, 2, and 3 coordinates) does not exhibit either shear-extension or shear-shear coupling. Recall that an orthotropic material has nine independent constants because... 

Thus, three reciprocal relations must be satisfied for an orthotropic material. Moreover, only 2, V13, and V23 need be further considered because V21, V31, and V32 can be expressed in terms of the first-mentioned group of Poisson s ratios and the Young s moduli. The latter group of Poisson s ratios should not be forgotten, however, because for some tests they are what is actually measured. 

Because the stiffness and compliance matrices are mutually inverse, it follows by matrix algebra that their components are related as follows for orthotropic materials ... 

The preceding restrictions on engineering constants for orthotropic materials are used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model. For boron-epoxy composite materials, Dickerson and DiMartino [2-3] measured Poisson s ratios as high as 1.97 for the negative of the strain in the 2-direction over the strain in the 1-direction due to loading in the 1-direction (v 2)- The reported values of the Young s moduli for the two directions are E = 11.86 x 10 psi (81.77 GPa) and E2 = 1.33x10 psi (9.17 GPa). Thus,... 

Show that the determinant inequality In Equation (2.48) tor orthotropic materials correctly reduces to v< 1/2 for isotropic materials. 

STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL... 

For orthotropic materials, imposing a state of plane stress results in implied out-of-plane strains of... 

The term Cg3 is zero because no shear-extension coupling exists for an orthotropic lamina in principal material coordinates. For the orthotropic lamina, the Qn are... 

The preceding isotropic relations can be obtained either from the orthotropic relations by equating E. to E2 and G. 2 to G or by the same manner as the orthotropic relations were obtained. 

A so-called specially orthotropic lamina is an orthotropic lamina whose principal material axes are aligned with the natural body axes ... 

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

The only advantage associated with generally orthotropic laminae as opposed to anisotropic laminae is that generally orthotropic laminae are easier to characterize experimentally. However, if we do not realize that principal material axes exist, then a generally orthotropic lamina is indistinguishable from an anisotropic lamina. That is, we cannot take away the inherent orthotropic character of a lamina, but we cpn orient the lamina in such a manner as to make that character quite difficult to recognize. 

Because of the presence of Q g and Q2e in Equation (2.84) and of 3 g and 326 f Equation (2.87), the solution of problems involving so-called generally orthotropic laminae is more difficult than problems with so-called specially orthotropic laminae. That is, shear-extension coupling complicates the solution of practical problems. As a matter of fact, there... 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

The apparent anisotropic moduli for an orthotropic lamina stressed at an angle 6 to the principal material directions vary with 6 as in Equation... 

The values in Figures 2-11 and 2-12 are not entirely typical of all composite materials. For example, follow the hints in Exercise 2.6.7 to demonstrate that E can actually exceed both E., and E2 for some orthotropic laminae. Similarly, E, can be shown to be smaller than both E. and E2 (note that for boron-epoxy in Figure 2-12 E, is slightly smaller than E2 in the neighborhood of 6 = 60°). These results were summarized by Jones [2-6] as a simple theorem the extremum (largest and smallest) material properties do not necessarily occur in principal material coordinates. The moduli Gxy xy xyx exhibit similar peculiarities within the scope of Equation (2.97). Nothing should, therefore, be taken for granted with a new composite material its moduli as a function of 6 must be examined to truly understand its character. 

In summary, the engineering constants for anisotropic materials and orthotropic materials loaded in non-principal material coordinates can be most effectively thought of In strictly functional terms ... 

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