Lamina orthotropic

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... 

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. 

The actual invariants in invariant properties of a lamina include not only U., U4, and U5 because they are the constant terms in Equation (2.93) but functions related to U-), U4, and U5 as shown in Problem Set 2.7. The terms U2 and U3 are not invariants. The only invariants of an orthotropic lamina can be shown to be... 

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. 

That the principal stresses are not of interest in determining the strength of an orthotropic lamina is illustrated with the following example. Consider the lamina with unidirectionai fibers shown in Figure 2-16. Say that the hypotheticai strengths of the lamina in the 1-2 piane are... 

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. 

Derive the summation expressions for extensional, bending-extension coupling, and bending stiffnesses for laminates with constant properties in each orthotropic lamina that is, derive Equation (4.24) from Equations (4.20) and (4.21). 

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... 

The stiffnesses of an antisymmetric laminate of anisotropic laminae do not simplify from those presented in Equations (4.22) and (4.23). However, as a consequence of antisymmetry of material properties of generally orthotropic laminae, but symmetry of their thicknesses, the shear-extension coupling stiffness A.,6,... 

The bending-extension coupling stiffnesses, Bjj, vary for different classes of antisymmetric laminates of generally orthotropic laminae, and, in fact, no general representation exists other than in the following force and moment resultants ... 

An antisymmetric cross-ply laminate consists of an even number of orthotropic laminae laid on each other with principal material directions alternating at 0° and 90° to the laminate axes as in the simple example of Figure 4-19. A more complicated example is given in Table 4-4 (where the adjacent layers do not always have the sequence 0°, then 90°, then 0°, etc.). Such laminates do not have A g, Agg, D g, and Dgg, but do have bending-extension coupling. We will show later that the coupling is such that the force and moment resultants are... 

For plane stress on an orthotropic lamina in principal material coordinates. 

Figure 4-37 Thermal Expansion and Distortion of an Orthotropic Lamina...

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

Consider an angle-ply laminate composed of orthotropic laminae that are symmetrically arranged about the middle surface as shown in Figure 4-48. Because of the symmetry of both material properties and geometry, there is no coupling between bending and extension. That is, the laminate in Figure 4-48 can be subjected to and will only extend in the x-direction and contract in the y- and z-directions, but will not bend. 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

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. 

Indeed, the slope of Figure 6-14 is actually -.49, so the theory is apparently applicable to an orthotropic lamina with cracks in the fiber direction. The contention is further substantiated by tests for the other loading paths shown in Figure 6-13. 

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 basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as... 

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... 

When all orthotropic laminae are of the same material, the constants U, U2, and Ug can be brought outside the integrals ... 

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