Higher-order bending elasticity

Fluid membranes are molecularly thin two-dimensional objects in three-dimensional space. Their extreme deformability distinguishes them from liquid crystals and other three-dimensional systems. Moreover, their fluidity and complex internal structure sets them apart from polymers. The intervention of higher order bending elasticity and its consequences for membrane shape and fluctuations seem to be related to these characteristic properties. 

Since in second-order expansion all terms are preceded by the constant factor S, the Li are order-parameter independent, and all elastic constants are proportional to the square of the order parameter. There are only two invariants in this expansion, and hence this approximation cannot describe correctly the physical situation but leads to equal values for the splay and bend constants. An expansion of the free energy to higher order terms in first derivatives has been performed by, among others, Schiele and Trimper [289], Berreman and Meiboom [290], Poniewierski and Sluckin [291] and Monselesan and Trebin [292]. 

The coefficient B is the elastic constant associated with the layer compressions. This represents the solid-like term along the layer normal. The second term (nematic-like) describes how much energy is required to bend the layers. In describing the SmA phase, both elastic constants K and B are very important. The former, which is higher order. 

In order that the bipartite beam should attain the higher bending strength as compared to a glulam beam of the same size and wood quality, the theoretical models indicate that the adhesive layers must exhibit a very special load-bearing behaviour. The adhesive manufacturer was able to produce some adhesive layers (like 027-2 in Fig. 7) which attained the requisite load-bearing behaviour for the ideal elastic adhesive layer. However, he was not quite able to fully attain the behaviour required for the ideal plastic adhesive layer. We decided to perform further tests with two selected adhesive layers (009-05 and 13-1 in Fig. 7), which came close to the desired performance needed for the ideal-plastic adhesive layer. There was a need to estimate the performance of bipartite beams with these adhesive connections. A programme based on the Excel solver function was developed to calculate the beam behaviour for these and other adhesive layers as follows. 

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