Stress-strains state calculation of a rod at uniaxial tension with non-local effects

A numerical calculation of a uniaxial extension of a rod is carried out in terms of the nonlocal integral theory of elasticity using the finite element method for various parameters of the material, taking into account its microstructure, and contributions of nonlocal effects. The dependencies of the strain energy are presented with increasing number of finite elements, which confirm the convergence and correctness of the mathematical model and method.


Introduction
The creation of new structurally sensitive materials based on nanotechnology is an important direction in the development of modern materials science. Consolidated structurally sensitive materials are usually obtained by compacting nanopowders, deposition on the substrate, crystallization of amorphous alloys and other methods.
Such materials have unique physicomechanical properties that allow them to be used effectively in structures exposed to high-intensity external influences [1,2].
An important step in the creation and usage of new structurally sensitive materials is the construction of mathematical models that allow describing their behavior in a wide range of changes in external influences.
It is known that the usage of methods of continuum mechanics for materials with a microstructure is limited to scale and boundary effects. Direct application of continuum mechanics methods for materials modified by micro-and nanostructured inclusions is incorrect [5, 14 -16]. For these reasons, there is an interesting theory where, on the one hand, is taken into account the presence of the microstructure, and on the other hand the equations have the form of the usual equations of continuum mechanics, integro-differential in the general case, for solutions which are applied the methods of continuum mechanics. Approaches of classical continuum mechanics media with micro-and nanostructure is called the method of continuous approximation [14,15]. The field of science in which the behavior of materials with micro-and nanostructures is studied using the method of continuous approximation is sometimes called The key points in this method are the establishment of connections between the characteristics of the micro (nano-) level and the macro level, as well as taking into account the effects of spatial and temporal non-locality of the media.

Mathematical model
Let us consider a mathematical model of a uniaxial tension of a rod ( fig. 1) with a length L, a cross-sectional area S and a concentrated force on the P right-hand side. The formulation of the boundary value problem of the mechanics of a deformed body in terms of small deformations has the form [11]: where b(x) is the force distributed over the volume, σ(x) and u(x) are the stress and displacement along the axis Ox. However, according to the non-local theory of elasticity [12], the relation between stresses and strains can be represented as where ε(x) -the strains along the axis Ox, E -the Young modulus, p 1 and p 2 are the proportions of the influence of local and non-local effects such as p 1 + p 2 = 1; ϕ(|x − x|) is an influence function that determines spatial nonlocality, while The influence function was used by A.C. Eringen [4,5,14,15] to solve problems in the theory of elasticity and based on the idea that long-range forces that responsible for the nonlocal deformation of the material at a given point of the space x are adequately described using the decreasing distance function ϕ(|x − x|), while increasing |x − x|.
As shown in [18], problem (1) can be reformulated into the Fredholm integral equation of the second kind with respect to strains, which does not have a strict analytical solution in the general case.

Nonlocal FEM-formulation
From the varational approach, the solution of problem (1) is the equivalent to minimizing a functional, which, in its physical sense, is the total potential energy of a loaded rod [11]: where V is the volume of the rod and S is the cross-sectional area of the right side. Using the discretization procedure, this problem can be solved in terms of isoparametric finite elements in displacements [13], then the original functional (3) can be rewritten in view of the relations in the form of FEM where N e is the number of elements, u n and u m are the displacement field in the element with number n and m, K loc n is the stiffness matrix of the element with the number n in terms of the classical theory of elasticity, K nl nm is matrix of non-local effects [27] that represents the influence of the m-th element on the n-th one ( fig. 2), f  (2) can be represented as where N n is the matrix of shape functions on the element with number n, B n and B m are the matrices of connection strains and stresses of the elements with numbers n and m, containing derivatives of the shape functions, V n is the length of the element with number n.
x O Like the matrix K loc n , the matrix of nonlocal effects K loc nm is calculated with isoparametric relations taking into account where ξ and η is local coordinates, J is the Jacobi matrix of the derivatives connection in the local coordinate system with the derivatives in global coordinate system, while B n = B n (x(ξ)) and B m = B m (x (η)). Similarly with the classical theory of elasticity, as shown in [13], the strain energy (7) in terms of the non-local theory of elasticity is finite in its physical meaning At the same time, turning to a finite element formulation, the strain energy should tend to its limit as the number of elements increases and the number of their characteristic size decreases, that is lim where W h is strain energy, computing at the current amount of elements, which can be represented as h is the maximum characteristic size of a finite element, which in the one-dimensional case is the length of the element.

Numerical results
The numerical experiment was carried out without influence of volume forces and with the following values of the material parameters: L = 100 mm, S = 10 mm 2 , E = 2.1 · 10 6 MPa and P = 10 N. The influence function selected for calculations was looked like where a is the characteristic of the material, which describes its microstructure. To calculate the matrices K loc n and K nl n numerical integration with Gauss quadratures over two points was used. Each calculation used linear finite elements of the same length in the amount of 100 elements.    As can be seen from the plots of the strain energy depending on the number of elements for various values of parameter a and various values of parameter of non-local effects p 1 (fig. 6), there is a convergence of numerical results confirmed.

Conclusion
Calculations of the stress-strain state of the rod at uniaxial tension, with non-local effects taking into account, showed: 1) in terms of the non-local integral theory of elasticity, a numerical solution obtained by the finite element method tends to its limit; 2) by increasing the value of parameter a, which determines the microstructure of the rod's material, leads to increasing strains in the zone of growth near the sides of the rod with increasing the maximum values of the strains; 3) by increasing the value of parameter a, the maximum displacement of the rod grows slightly; 4) increasing the value of parameter a, leads to increasing strains drop in the region of ToPME IOP Conf. Series: Materials Science and Engineering 489 (2019) 012025 IOP Publishing doi:10.1088/1757-899X/489/1/012025 7 the sides of the rod while decreasing the maximum stress values; 5) an increase in the share of the contribution of non-local effects does not have a significant effect on the strain energy of the entire rod.

Acknowledgements
The reaserch was supported by the Russian Federation for Basic Research project No. 18-38-20108.