Departamento de Gravitación y Teoría de Campos

                          Seminarios 2016

  

                          Instituto de Ciencias Nucleares, UNAM
[ ICN-UNAM]

Jueves 21 de abril, a las 17:00 hrs en el salón de seminarios A2.25, 2. piso del ICN

Jemal Guven (ICN, UNAM)

The rise and fall of free masonry arches

In equilibrium, the pull of gravity on the mass of an arch, a vault or a dome sets up stresses to oppose it. If the wall is thin the low energy modes of deformation will be surface isometries. But isometry is a geometrical constraint; as a consequence, the Euler-Lagrange equations describing equilibrium involve only geometrical degrees of freedom. In masonry the stresses ideally are compressive but the optimal design itself is largely independent of the material used to build it. Curiously, resistance to bending--never mind stretching or shear--plays no direct role in the design; neither does it contribute to the stress distributed within the structure. A simple catenary arch (or vault) illustrates the point. This arch would, of course, be very unstable with respect to small deformations were it not for its resistance to bending. The relevant energy is quadratic in curvature deviations away from the designed equilibrium. This offset or spontaneous curvature is typically both inhomogeneous and anisotropic. The simplest model capturing the physics involves a single parameter $\mu$ characterizing the bending rigidity of the medium. Intuitively $\mu$ must exceed a critical value if the arch is to stand. Accessing this stability involves expanding the complete energy--gravitational potential plus bending--to second order in deformations and the identification of the linear self-adjoint partial differential operator ${\cal L}$ controlling the response. The technical hitch is the accommodation of isometry. We examine the functional dependence of the spectrum of ${\cal L}$ on $\mu$ for a catenary arch. In particular, for each aspect ratio, we identify the critical rigidity required to stabilize the arch. The number of unstable modes, as well as the symmetry of the most unstable mode depends on how far $\mu$ lies below this critical value; when it is just sub-critical, there are two degenerate unstable modes, one raising the arch at its center, the other lowering it. By examining higher order corrections the latter is identified as the dominant instability of the arch.