Orientation orthotropous

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. 

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. 

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 ... 

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. 

Then, obviously the maximum principal stress is lower than the largest strength. However, 02 is greater than Y, so the lamina must fail under the imposed stresses (perhaps by cracking parallel to the fibers, but not necessarily). The key observation is that strength is a function of orientation of stresses relative to the principal material coordinates of an orthotropic lamina. In contrast, for an isotropic material, strength is independent of material orientation relative to the imposed stresses (the isotropic material has no orientation). 

Now that the basic stiffnesses and strengths have been defined for the principal material coordinates, we can proceed to determine how an orthotropic lamina behaves under biaxial stress states in Section 2.9. There, we must combine the information in principal material coordinates in order to define the stiffness and strength of a lamina at arbitrary orientations under arbitrary biaxial stress states. 

Demonstrate that the force per unit width on a two-layered laminate with orthotropic laminae of equal thickness oriented at -h a and - a to the applied force is... 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

For cross-ply laminates, a knee in the load-deformation cun/e occurs after the mechanical and thermal interactions between layers uncouple because of failure (which might be only degradation, not necessarily fracture) of a lamina. The mechanical interactions are caused by Poisson effects and/or shear-extension coupling. The thermal interactions are caused by different coefficients of thermal expansion in different layers because of different angular orientations of the layers (even though the orthotropic materials in each lamina are the same). The interactions are disrupted if the layers in a laminate separate. 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

The invariant stiffness concepts for a iamina will now be extended to a laminate. All results in this and succeeding subsections on invariant laminate stiffnesses were obtained by Tsai and Pagano [7-16 and 7-17]. The laminate is composed of orthotropic laminae with arbitrary orientations and thicknesses. The stiffnesses of the laminate in the x-y plane can be written in the usual manner as... 

Anisotropic material In an anisotropic material the properties vary, depending on the direction in which they are measured. There are various degrees of anisotropy, using different terms such as orthotropic or unidirectional, bidirectional, heterogeneous, and so on (Fig. 3-19). For example, cast plastics or metals tend to be reasonably isotropic. However, plastics that are extruded, injection molded, and rolled plastics and metals tend to develop an orientation in the processing flow direction (machined direction). Thus, they have different properties in the machine and transverse directions, particularly in the case of extruded or rolled materials (plastics, steels, etc.). 

Orientation of reinforcement The behavior of RPs is dominated by the arrangement and the interaction of the stiff, strong fibers with the less stiff, weaker plastic matrix. The features of the structure and the construction determine the behavior of RPs that is important to the designer. A major advantage is the fact that directional properties can be maximized in the plane of the sheet. As shown in Fig. 8-55 they can be isotropic, orthotropic, etc. Basic design theories of combining actions of plastics and reinforcements... 

A few examples of the moduli of systems with simple symmetry will be discussed. Figure 1A illustrates one type of anisotropic system, known as uniaxial orthotropic. The lines in the figure could represent oriented segments of polymer chains, or they could be fibers in a composite material. This uniaxially oriented system has five independent elastic moduli if the lines (or fibers) ara randomly spaced when viewed from the end. Uniaxial systems have six moduli if the ends of the fibers arc packed in a pattern such as cubic or hexagonal packing. The five engineering moduli are il-... 

J. S. Cintra and C. L. Tucker, Orthotropic closuere approximations for flow-induced fiber orientation, J. Rheol. 39, 1095-1122 (1995) C. V. Chaubal, A. Srinivasan, O. Egecioglu, and L. G. Leal, Smoothed particle hydrodynamics technique for the solution of kinetic theory problems. Part 1. Method, J. Non-Newtonian Fluid Mech. 70, 125-54 (1997). 

The polymer sample is assumed to have at least orthotropic symmetry, i.e. it contains three mutually perpendicular directions such that if it is rotated through 180° about any one of these directions its macroscopic properties are unchanged. Axes OZ1Z2A3 are chosen parallel to these three symmetry directions of the sample. The orientation of a particular structural unit can then be specified in terms of three Euler angles, 6, and jr, as shown in fig. 10.7. 

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,... 

Figure 8.26 Composite panel with layers oand 6of different orthotropic materials oriented at arbitrary angles a and p with respect to applied stresses o, 02, and T/p.

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