Modeling the deformation process of a plate with a stress concentrator taking into account the postcritical stage of material deformation

Postcritical deformation of a material is a process which is characterized by a decrease of stress during growing deformations as a result of accumulation of structural damage. The design becomes unable to withstand the external load only when zones with weakened connections are developed enough. Evolution of postcritical deformation zones can occur with an increase of the external load applied to the construction. It means that taking into account the softening of the material allows determining the strength and deformation reserves of constructions more accurately. The mathematical formulation of the boundary value problem of supercritical deformation mechanics is given in the paper. The features of the experimental study of the postcritical stage of material deformation are listed. Strain curves of various steels with a long section of softening are obtained. Numerical solutions for the problems of deforming a thin plate with stress concentrators of different geometries under kinematic loading are obtained. Piecewise linear approximations and real strain curves of steel 20 and steel 40Cr4 obtained experimentally are considered. The evolution of zones of postcritical deformation in the material is considered. The correspondence between the value of the decline modulus and the nature of the evolution of the softening zones is determined. A stress plot is constructed that reflects how the complete material deformation diagram is realized near the concentrator. The calculated loading diagrams are constructed. It is noted that even after the appearance of softening zones, an increase in external load is possible. The strength and deformation resources of structures are determined, and the influence of the geometry of the stress concentrator on their values is considered. It is noted that the consideration of softening in modeling the behavior of structures with stress concentrators is appropriate.

NONLINEAR BEHAVIOR CALCULATION ALGORITHM FOR THIN-WALLED SYSTEMS

Objectives The emergence of modern high-strength materials leads to the creation of thin-walled structures in various fields of technology. To obtain the necessary information about their behavior under load, one should analyze all the characteristic features encountered at all stages of their loading - at the initial (initial) stage of their operation, taking into account one or more types of nonlinearities, find possible critical states and, depending on the type of stability loss, study the nature of the initial stage of postcritical deformation. Based on an algorithm combining approximate analytical and numerical methods, the article solves the model problem — studying the behavior of a thin-walled spherical shell under load.Method. The study is based on solving the nonlinear problem of determining the stress-strain state at the initial - axisymmetric stage of work; critical (bifurcation) load values; analysis of the nature of post-bifurcation behavior. The work uses a variant of the general theory of stability and postcritical behavior of structures previously developed by V.T. Coiter.Result. The solution of such a general problem associated with discontinuous phenomena is carried out on the basis of mathematical ideas formulated in the theory of branching solutions of nonlinear equations. The values of the coefficients characterizing the initial stage of the post-bifurcation behavior of the shells and, from a practical point of view, the relations between the critical and limiting values of the loads are obtained. It is shown that depending on the area of the shell surface part loaded by the distributed load, the nature of the initial stage of postcritical deformation changes not only quantitatively, but also qualitatively.Conclusion. The most effective in solving problems associated with discontinuous phenomena are combinations of approximate analytical ones - catastrophe theory and numerical methods that do not require complex, timeconsuming and significant amounts of computation. Analysis of the initial stage of the postbifurcation behavior of structures allows us to assess the degree of danger of reaching a critical state, which is achieved by taking into account the values of the corresponding reliability coefficients in the calculations.