Axisymmetric problem within strain-gradient elasticity and mixed FEM solution for cylinders with cracks under tension

The two-dimensional axisymmetric boundary value problem was formulated in cylindrical coordinates within simplified strain-gradient elasticity. A mixed variation formulation of the finite element method (FEM) was applied to solve the set of modeling axisymmetric boundary value problems of solid and hollow cylinders of finite length with cracks, namely, the axial tension of the solid cylinder with disk and external edge circular crack and the hollow cylinder with inner and external edge circular crack. This mixed approximation for displacements, deformations, stresses, and their gradients has a principal difference from others available in the literature. It ensures the continuity of displacement, strain, and stress fields throughout the area occupied by the body and satisfies the necessary and sufficient condition, which provides the uniqueness of the solution and the stability of this approximation within strain-gradient elasticity theory. Crack-opening displacements as a function of distance from the crack center, and radial, axial, and normal circumferential stresses along the radius are presented in diagrams for different scale parameter values. All problems demonstrate specific qualitative features characterizing strain-gradient solutions: increasing crack stiffness with length scale parameters, cusp-like closure effect, and finiteness of Cauchy stresses at the crack tip. To the best of authors’ knowledge, such problems are considered within strain-gradient elasticity theory for the first time. Only the one-dimensional problem of a hollow infinite cylinder under internal and external hydrostatic pressure (loading does not depend on the polar angle), solved in a polar coordinate system, is available in the literature.