In the present study, a comparison of pure, three-point, four-point and cantilever beam bending problems in the frame of the theory of elastic materials with voids (micro-dilatational elasticity) has been provided via analytical modelling and three-dimensional finite-element analysis. We consider the extended variant of the theory with surface effects using a variational approach. At first, we compare the known approximate semi-inverse analytical solution of the pure bending problem with a corresponding three-dimensional finite-element solution in the frame of micro-dilatational theory. It is demonstrated that in the numerical solution – unlike the analytical one – all boundary conditions are satisfied accurately, and there exist distortions of the cross-sections and lateral faces of the beam. The generalized analytical solution of the beam pure bending problem with surface effects is also established and compared with numerical simulations. The effective elastic properties of the beam with micro-dilatations are introduced by comparing its displacements and corresponding classical beam displacements. The influence of scale, coupling and surface parameters on the effective elastic moduli in different bending and simple tension tests is studied. It is shown that all considered types of bending experiments provide the determination of close values of the effective flexural modulus of the beams with different thickness. This means that it is possible to use any bending test with beams of different thickness for reliable identification of the material constants of the theory. It is also possible to use the simple analytical expression for an effective flexural modulus that follows from the analytical solution of the pure bending problem. It is also shown that stiffness of the beam with micro-dilatation in any bending tests should always be higher as compared with the stiffness of such a beam in simple tension. For thin beams, there are no scale effects, and its effective flexural modulus is equal to the material’s Young’s modulus without micro-dilatation. Possible rise of the negative size effects in the model with surface effects is also discussed.

A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich-Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich-Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov-Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments.

Eshelby integral formulas play a fundamental role in mechanics of composite materials, because they provide an efficient tool for determining the average properties of dispersion-filled materials. For example, their use in the framework of the self-consistent averaging method actually gives a final and quite precise solution to the problem of determining effective physical and mechanical properties of filled composites up to large relative contents of inclusions and almost all relations between the phase characteristics of the composite. In the present paper, we generalize the Eshelby integral formulas to the gradient theory of elasticity. This provides the possibility for using efficient methods for estimating the average characteristics of micro and nano-structured materials in the framework of gradient theories, which permit taking the scale effects into account correctly, and hence find wider and wider applications in describing the mechanical and physical processes.