The Mysterious Shapes Formed by Cracks

When a solid object fractures and breaks apart, numerous cracks appear. These cracks connect, dividing the original solid into several smaller fragments. When a stone hits a window, ordinary glass shatters into polygonal fragments of a certain size. Tempered glass, on the other hand, either develops a fine mesh of cracks to prevent shattering or breaks into small, granular pieces that cascade down. Similarly, striking a rock with a hammer breaks it into fragments of various sizes. In all these cases, our intuition simply tells us that things break apart, and we see nothing particularly mysterious in it.

But what about the tortoise-shell-like cracks on the surface of a dried rice paddy? I am sure everyone has seen them at least once. Why does the surface divide into such polygonal cells? What determines the size of these cells? Now that you mention it, don't they seem to be a rather mysterious shape? (Figure 1)

An even more mysterious shape than the cracks on a paddy field's surface is "columnar jointing." An internet search for "columnar jointing" reveals many famous and scenic wonders in Japan and around the world, such as Tojinbo, the Giant's Causeway, and Devils Tower. Columnar jointing is a geological structure in which cracks form in a massive rock, dividing it into numerous polygonal columns that look like a bundle of pencils standing upright. When viewed from above, these columnar joints appear as an array of polygonal cells, much like the cracks on the surface of a rice paddy. (Figure 2)

I use numerical analysis methods to study the fracture of solids. Within the field of fracture research, the mysterious shapes formed by cracks are a fascinating subject. The cracks in rice paddies and columnar jointing are "multiphysics" problems, where multiple physical phenomena progress while mutually influencing one another. "Mysterious shapes" like these are often formed as a result of such multiphysics, where several physical phenomena are intertwined. Specifically, the cracks in a rice paddy represent a coupled problem of three phenomena: the diffusion that governs moisture transfer, the object's deformation due to moisture expansion and drying shrinkage, and the fracture that results from advanced deformation. This is a coupled problem of "diffusion, deformation, and fracture." Columnar jointing is also a coupled problem of "diffusion, deformation, and fracture," involving the diffusion that governs heat transfer as lava cools rapidly, and the object's deformation and fracture due to thermal expansion and contraction. When analyzing these coupled problems, the treatment of "fracture" is particularly challenging. A crack resulting from fracture is a discontinuous field that suddenly appears within a continuous deformation field. Handling such fields is difficult in numerical analysis. Rice paddy cracks and columnar jointing are problems of "pattern formation in multiphysics involving the evolution of a discontinuous field," and they represent one of the most difficult challenges in numerical analysis. To date, I have successfully performed a numerical analysis of rice paddy cracks (Figure 3). My next target is columnar jointing.