Abstract
Mathematical models of planar physically nonlinear inhomogeneous plates with rectangular cuts are constructed based on the three-dimensional (3D) theory of elasticity, the Mises plasticity criterion, and Birger’s method of variable parameters. The theory is developed for arbitrary deformation diagrams, boundary conditions, transverse loads, and material inhomogeneities. Additionally, inhomogeneities in the form of holes of any size and shape are considered. The finite element method is employed to solve the problem, and the convergence of this method is examined. Finally, based on numerical experiments, the influence of various inhomogeneities in the plates on their stress–strain states under the action of static mechanical loads is presented and discussed. Results show that these imbalances existing with the plate’s structure lead to increased plastic deformation.
Highlights
Much attention in modern science is paid to the study of the stability of plates and shells that have holes in their structures
Global and local studies of the stress distribution in the vicinity of a discontinuity in a structure like cutouts and cracks play a crucial role in designing plates and shells
We study the convergence of the solution of the finite element method (FEM) for a given problem with a uniformly distributed load q 11.25 for a plate with four holes (Fig. 4c)
Summary
Much attention in modern science is paid to the study of the stability of plates and shells that have holes in their structures. Dzhabrailov et al [17] derived the governing equations for the relationship between stress and strain increments under planar loading in the elastoplastic stage It was based on the theory of small elastic–plastic deformations, but the considered case studies showed the effectiveness of the developed algorithms for shells with plane load outside the elastic limit. Awrejcewicz et al [34] investigated geometrically nonlinear vibrations of shallow laminated shells with complex forms and different boundary conditions They used the R-function theory, variational Ritz method, the Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mathematical models of physically nonlinear inhomogeneous planar plates with rectangular cuts are built on the basis of the 3D theory of elasticity It is based on the deformation theory of plasticity, the Mises plasticity criterion and the method of variable parameters of Birger elasticity, and the convergence theorem devoted to this method for a physically nonlinear body in a 3D formulation.
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