Introduction to special issue on stability under finite deformation

In the botanical gardens of Grenoble, France, there is small bridge, just a few metres long, made of pre-stressed concrete. The bridge dates back to 1855 and is thought to be the oldest manufactured pre-stressed structure made with concrete. In general, a pre-stressed bridge span is reinforced by parallel metallic rods embedded below the horizontal mid-plane and put under tension, so that the slab is bent slightly upwards. When the bridge is put in place, the span is subjected to its own weight and to external loads, which ensure that the concrete is compressed everywhere. Without the prestress, the weight and loads would bend the span downwards, and create a zone of tension, of which concrete cannot sustain much. This technology thus combines the compressive strength of concrete with the tensile strength of the metallic rods. We can say that the resulting structure is stabilized by prestress. Of course, a prestress or a prestrain can conversely lead to the destabilization of a solid. A classic example is that of the twisting instability of a closed ring: take an elastic straight rod, bend it into a circle by bringing the two ends together, but give the rod a twist before gluing its ends. For a sufficiently high twist, the elastic ring is unstable and folds unto itself, by forming an eight-shape curve (with about two turns of pretwist) or more elaborate shapes (for higher twists), see Fig. 1. Stability studies are highly important for many engineering materials, be they subjected to small prestrains such as those imposed on pre-stressed concrete, as well as to finite prestrains, such as those imposed on elastomers in bridge bearings and engine mountings. These studies have developed into a fundamental topic of mathematical modelling because they are relevant not only to engineering problems but also to many biological, biomedical and biomechanical applications (DNA mechanics, cell stiffness, cellular structures, plant growth, microbial filaments, deformation of arteries and veins, skin wrinkling, etc.). The discipline of elastic stability has a long and distinguished history, dating back at least to the works of Euler. For a comprehensive bibliography and an extensive treatment of most known problems, we refer, e.g. to the textbook by Bažant & Cedolin (2003). For this introduction to the Special Issue on ‘Stability under Finite Deformations’, we chose to evoke three seminal papers, which we believe qualify as pioneering works in this field, and yet seem to be little known and forgotten (to the best of our knowledge). First, an article by Louis-Augustin Cauchy (1829), which contains the first derivation of the equations of motion in a solid which is already in a state of stress. According to Truesdell (1966), Cauchy’s ‘results were not understood and were reported obscurely or even incorrectly by 19th century