A survey of A. Yu. Ishlinsky’s papers on the occasion of his 100th birthday

Abstract

A. Yu. Ishlinsky’s scientific publications impress one by the extremely wide scope of his scientific interests. Most of his results lie in two fundamental fields of research, mechanics of deformable bodies and mechanics of gyroscope and navigation systems; chronologically, they appeared mainly in this order. One of his first important results that became widely known was the construction of an exact theory of the Brinell test (1942). This theory dealt with determining the stress state under a rigid punch of spherical or plane shape penetrating into a perfectly plastic medium. The problem was solved in the rigorous statement in the framework of the Haar–von Karman complete plasticity hypothesis containing two relations between the principal stresses rather than one relation as in the Saint-Venant theory. The subsequent development of this problem led (1954) to construction of a general theory of plasticity which, in particular, allowed one to overcome the difficulties related to the derivation of the 3D strain equations beyond the elasticity range. A series of Ishlinsky’s papers (1940–1946) deals with imperfect elasticity, fracture theory, and plasticity theory. In this series, he constructed a description of aftereffect and relaxation generalizing the Maxwell and Thomson concepts on the basis of a linear differential relation between the stress and the strain. Plastic fracture models were also studied in these works. In particular, one of the fracture models was constructed for the example of a body whose one-dimensional strain obeys the law dσ/dt+ rσ= bde/dt+ bne, where σ is the stress and e is the strain. This model also admits a simple and illustrative interpretation as a damper with springs, which permits readily understanding the introduced fracture hypotheses and analyzing specific examples for various loading laws. Ishlinsky returned to plasticity and fracture theories in 1965–1968. The problem on the impact of a viscoplastic rod on a rigid obstacle is of interest. In the model accepted by Ishlinsky, the motion of the rod cross-sections is governed by heat equations with complex boundary conditions. It turned out unexpectedly that on impact the viscoplastic state does not instantly extend to the entire rod; instead, it runs away from the impact point and then returns without reaching the free end, thus leaving a rigid segment of the rod. In fracture theory of this time period, one should mention Ishlinsky’s paper of 1968, where two crack development models are compared. One model is the Griffith–Irvin model, and the other, which was developed later, is due to S. A. Khristianovich and G. I. Barenblatt. Ishlinsky showed that the latter model tends to the former as the range of the adhesion forces postulated in the latter tends to zero. Ishlinsky’s papers on the stability of elastic systems are of extreme importance. In a joint 1949 paper with M. A. Lavrent’ev, he theoretically explained the spectacular experiment in which a tube subjected to an underwater explosion loses its shape in such a way that the number of corrugations is the greater the closer they are to the explosion site. It turned out that the relationship between the characteristic exponents, the equilibrium mode number, and the applied external load constructed when studying the equilibrium modes permits one to maximize the buckling rate with respect to the mode number. The greater the applied force, the higher the mode number maximizing the characteristic exponent is. This result has numerous practical applications. The buckling problem for elastic systems, which goes back to Euler, is usually solved by methods of the theory of strength of materials for simplified models of rod or plate type. Ishlinsky studied this