An incremental hybrid natural element method (HNEM) is proposed to solve the two-dimensional elasto-plastic problems in the paper. The corresponding formulae of this method are obtained by consolidating the hybrid stress element and the incremental Hellinger-Reissner variational principle into the NEM. Using this method, the stress and displacement variables at each node can be directly obtained after the stress and displacement interpolation functions are properly constructed. The numerical examples are given to show the advantages of the proposed algorithm of the HNEM, and the solutions for the elasto-plastic problems are better than those of the NEM. In addition, the performance of the proposed algorithm is better than the recover stress method using moving least square interpolation.
The natural element method (NEM) [
The elastoplasticity problem is an important nonlinear problem in solid mechanics. Based on the NEM, some research work to solve the elastoplastic problems has been implemented over the last twenty years. Toi and Kang [
Due to the
By introducing the hybrid stress element into the natural element method and combining the incremental Hellinger-Reissner variational principle [
For the region
Discrete model of region
Suppose
The 2nd order Voronoi cell about
During the process of problem-solving, the natural neighbor nodes of
The interpolating Sibson base function can be written as
For mechanics problem we can use the following to be displacement interpolation function:
Because the shape functions of natural neighbor nodes are
For two-dimensional problem, the displacement, strain, and stress fields of a body are defined to be
The equilibrium equation can be written as
The strain-displacement relationship can be written as
The stress-strain relationship can be expressed as
During elasticity stage,
Within plastic stage
where
The boundary conditions can be written as
Let us suppose that there exist
Equations of (
Applying the incremental Hellinger-Reissner variation principle given by [
Among above
Suppose the displacement boundary condition (
According to stationary value condition of multivariation increment variation principle,
We have
Substituting (
Based on the stationary condition of variation,
The equations for solving generalized displacement can be found:
For the elastoplasticity problem with small deformation, the relationship of stress and strain in the plastic field is nonlinear. The Newton-Raphson iterative solver is used to solve nonlinear system. We convert the nonlinear system into a series of linear problems using the method of gradually increasing loading. After the structure enters the yield state, the load is added with load increment method. The structure is added with the load increment
For the solution of (
Figure
Cantilever beam subjected to a concentrated force.
Figure
The vertical displacement of the neutral axis of beam.
Relationship between displacements of the midpoint at the end of the beam and the load.
Tables
The normal stress
Coordinate of |
Analytical | HNEM | NEM |
---|---|---|---|
(4.267, −0.292) | −13.067 | −12.988 | −10.076 |
(5.333, −0.458) | −14.667 | −14.055 | −12.288 |
(5.876, 0.208) | 5.333 | 5.203 | 4.092 |
(6.533, 0.042) | 0.733 | 0.779 | 0.413 |
(7.467, 0.488) | 2.933 | 2.764 | 2.111 |
The shear stress
Coordinate of Gaussian points | Analytical | HNEM | NEM |
---|---|---|---|
(4.133, 0.292) | −0.990 | −0.959 | −4.615 |
(5.067, 0.208) | −1.240 | −1.377 | −2.574 |
(6.533, 0.167) | −1.333 | −1.566 | −2.488 |
(6.933, 0.042) | −1.490 | −1.376 | −0.824 |
(7.333, 0.167) | −1.333 | −1.166 | −1.704 |
Figure
A cylinder with uniform pressure on inner surface.
We can only consider one quarter of the cylinder because of the symmetry, as shown in Figure
A quarter of the cylinder with boundary conditions.
Node distribution and Delaunay triangles of a quarter of the cylinder.
The radial displacement at
Relationship between the radial displacements at the point (5.0, 0.0) and the loading.
A plate with a central hole is fixed at one end and subjected to axial uniform tension at the other end, as shown in Figure
The plate with a central hole under axial uniform tension.
The nodes layout of the plate is given in Figure
Node distribution and Delaunay triangles of plate.
The horizontal displacement at
The vertical displacement at
Relationship between the vertical displacements at the point (5.0, 0.0) and the loading.
Based on the natural element method (NEM), by introducing the hybrid stress element into the NEM and combining the incremental Hellinger-Reissner variational principle, the incremental hybrid natural element method (HNEM) has been proposed for solving two-dimensional elastoplastic problems in the paper. The corresponding formulae of this method are deduced and the relevant programs of this method are compiled. Three numerical examples of two-dimensional elastoplastic problems are given, showing that with the proposed algorithm of the HNEM, the more accurate solutions for the elastoplastic problems can be obtained by the HNEM than those of the NEM. In addition, the efficiency of the proposed algorithm has been improved compared with the recover stress method using moving least square interpolation.
Using the HNEM in this paper, the stress values and displacement values of nodes can be easily got at the same time by structuring suitable node stress function and displacement interpolation function. Then not only the advantages of the convenience in pre-process, the simple shape function, and it satisfies the Delta interpolation on the boundary of the NEM are reserved by the HNEM, but also the problem of not directly solve the stresses on nodes is resolved and the accuracy of stresses is improved.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work was supported by the Natural Science Foundation of Shanghai China (no. 13ZR1415900) and Program of Shanghai Science and technology Commission (no. 11231202700).