On the theory of stress concentration for shear‐strained prismatical bodies with a non‐linear stress‐strain law

Abstract

The system of equations governing the stress in a shear‐strained prismatical body are examined and it is shown that, provided the stress‐strain law adopts certain multi‐parameter forms, the system may be reduced to the Cauchy‐Riemann equations. Integration of the system is then immediate and the analysis of the stress concentration round certain notches thereby facilitated.

The Korteweg capillarity system. Integrable reduction via gauge and reciprocal links

“…Accordingly, the latter is here seen to be gauge equivalent to the canonical integrable cubic NLS equation in . Moreover, if, in addition, q c = 0 then (25) reduces to the Kundu-Eckhaus equation, linked to a linear Schrödinger equation in via gauge transformation (20). Here, we shall be concerned with the general case with q c = 0, q q = 0, s = 0 and alignment of (24) is made with the nonlinear capillarity model (19).…”

“…Accordingly, the latter is here seen to be gauge equivalent to the canonical integrable cubic NLS equation in . Moreover, if, in addition, q c = 0 then (25) reduces to the Kundu-Eckhaus equation, linked to a linear Schrödinger equation in via gauge transformation (20).…”

“…with σ, τ in the wave packet ansatz (28) then given by the relations (39) and the class of exact solutions of the quintic NLS equation (24) via the gauge transformation (20). Moreover, two-parameter solutions of (41) of the type…”

“…It has been seen that the action of the gauge transformation (20) is to inject a parameter γ into the class of solutions constructed by the algorithmic application of the invariant relations (31) and (32) involving the constants of motion I and H. This may be combined with the action of the reciprocal transformation (57) to inject, in addition, the parameter χ into the class of solutions of the Korteweg capillarity system with…”

“…It was much later established that a re-interpretation and generalisation of that work has remarkable solitonic connections whereby novel master systems of 2+1-dimensional integrable soliton equations may be derived [18,19]. In the area of nonlinear elasticity, the Bäcklund transformations of [12] were shown in [20] to be applicable to the classical elastostatic problem of determining the stress distribution in shear-strained, isotropic prismatical bodies with certain nonlinear model deformation laws. The stress and warping distributions were thereby calculated for a class of boundary indentation problems involving sharply curved notches, notably for the Neuber-Sokolovsky stress-deformation constitutive relations.…”