Theoretical stiffness

Ciassicai lamination theory is derived in Section 4.2. Then, special stiffnesses of practical interest are classified and examined in Section 4.3. Next, the theoretical stiffnesses obtained by classical lamination theory are compared with experimental results in Section 4.4. In Section 4.5, the strengths of various laminates are predicted. Finally, the stresses between the laminae of a laminate are examined in Section 4.6 and found to be a proba lei causg of delamination of some laminates. 

The group-theoretical stiffness parameters can be expressed in the conventional (cubic) elastic stiffness coefficients ... 

The initial stimulus for the preparation of high modulus polyethylenes came from the recognition of the gap between the available stiffness and the theoretical stiffness. As indicated in this review the gap has now been largely bridged, which is emphasised still further by consideration of the total mechanical anisotropy as shown in Table 5. 

Although it had been appreciated since the early studies of Meyer and Lotmar 11 in the 1930 s that there was a substantial gap between the theoretical stiffness of the chain in several commercially available polymers and the achievement of stiffness by existing processes, it was not until the 1970 s that this gap was bridged. It is in polyethylene that the results have been most dramatically realised, and oriented fibres and rods have been produced with room temperature Young s moduli in the range from 50-100 GPa 2-7). Solution spun fibres have even been prepared with a Young s modulus at low temperatures of 288 GPa 8), which is very close to the highest theoretical estimate of 324 GPa 9). 

In this paper, the Equivalent Constraint Model (ECM) of the damaged lamina was applied to predict theoretically stiffness-damage behaviour of cross-ply [0m/90 ] ... 

Theoretical analysis of certain features in the electromagnetic spectrum yields basic molecular parameters such as bond lengths and bond stiffness. We shall see presently that the mechanical spectra can be related to molecular parameters and not just modelistic characteristics as we have used until now. 

The theoretical and measured results for E, are shown in Figure 3-41 as a function of resin content by weight. Theoretical results from Equation (3.64) are shown for C = 0,. 2,. 4, and 1, and the data are bounded by the curves for C = 0 and C =. 4. The theoretical curve labeled glass-resin connected in series is a lower, lower bound than the C = 0 curve and is an overly conservative estimate of the stiffness. 

Two- and three-layered special cross-ply laminates were shown to have extrema of behavior in the preceding section. Thus, comparisons between theoretical and measured stiffnesses for such laminates should... 

The measured stiffnesses for two- and three-layered special cross-ply laminates are shown with symbols in Figure 4-28, and the theoretical results are shown with solid lines. In all cases, the load was kept so low that no strain exceeded SOOp. Thus, the behavior was linear and elastic. The agreement between theory and experiment is quite good. Both the qualitative and the quantitative aspects of the theory are verified. Thus, the capability to predict cross-ply laminate stiffnesses exists and is quite accurate. 

Figure 4-28a Theoretical and Measured Special Cross-Ply Laminate Stiffnesses (U. S. Standard Units) (After Tsai [4-6])...

The theoretical and measured stiffnesses are shown in Figure 4-32. As with cross-ply laminates, very good agreement was obtained. Thus, the predictions of laminate stiffnesses are quite accurate. 

Theoretical and measured strengths and stiffnesses of three-layer cross-ply laminates with cross-ply ratios ranging from. 2 to 4 are shown in Figure 4-44. The scatter in the data is partially due to the difficulty of making good tensile specimens the characteristic dog-bone shape is formed by routing that often damages the 90° layer. 

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