Configurational phases in elastic foams under lengthscale-free punching

We carry out experiments with brick-like specimens of elastic open-cell (EOC) foams of three relative densities. Individual specimens may be “tall” (height = width = depth) or “short” (2height=width=depth). We place each specimen on a supporting plate and use a lengthscale-free (wedge-shaped or conical) punch to apply forces downward along the specimen's height. Regardless of the type of specimen, the force–penetration curves remain linear, for the wedge-shaped punch, or quadratic, for the conical punch, up to a sizable penetration commensurate with the smallest lengthscale of the specimen. After that there is an abrupt, all-but-discontinuous change in stiffness: if the specimen is tall, the stiffness drops; if the specimen is short, the stiffness shoots up. To analyze these curious experimental results, we posit that EOC foams can be found in either of two configurational phases, here termed the low-strain phase and the high-strain phase, which share a two-dimensional interface (a surface of strain discontinuity). The analysis may be outlined as follows. In the first part of an experiment, there obtains a “similarity regime” in which the penetration of the punch and the radius of the interface are the only prevailing lengthscales (because the punch is lengthscale free). In this case, it is possible to show that the force–penetration curve must be linear, or quadratic, depending on whether the punch be wedge-shaped or conical, respectively. This prediction of the analysis is consistent with the experiments. In time, the similarity regime breaks down when the interface reaches one of the specimen's boundaries distal to the tip of the punch. If the specimen is tall, the soft, stress-free lateral boundary is reached first, and the stiffness must drop; if the specimen is short, the hard boundary in contact with the supporting plate is reached first, and the stiffness must shoot up. These predictions too are consistent with the experiments. To provide direct empirical evidence of the interface, we use a digital-image correlation method. Lastly, we run computational simulations of all the experiments, using finite elements and the skeleton-and-bubble model of EOC foams. The computational results are in good accord with the experimental ones, and they allow us to carry out a detailed validation of the analysis. Our findings evince the cardinal role of configurational phases in the mechanics of EOC foams.