TEST CASE 7.4 CUBE UNDER THE CONDITIONS OF CONSTANT STRESSES BY VOLUME
Reference:
R. H. Macneal, R. L. Harder, A proposed standard set of problems to test finite element accuracy, North-Holland, Finite elements in analysis and design, 1, 1985, p. 3-20.
Problem description:
A single isotropic cube is subjected to displacements of the outer surfaces, which provide conditions for constant stresses over the volume.
Problem sketch:
Type of created problem:
Spatial farm or volumetric array (X, Y, Z).
Geometric characteristics:
Cube edge: a = 1 m;
Coordinates of points: 1 (0.35, 0.35, 0.35); 2 (0.75, 0.25, 0.25); 3 (0.85, 0.85, 0.15);
4 (0.25, 0.75, 0.25); 5 (0.35, 0.35, 0.65); 6 (0.75, 0.25, 0.75); 7 (0.85, 0.85, 0.85); 8 (0.25, 0.75, 0.75).
Material properties:
Isotropic elastic: E = 1·10^{6} kPa; μ = 0.25.
Boundary conditions:
Absent — the immutability of the system is provided by loads.
Loads:
Node offset: u = 10^{-3}·(2·x^{ }+ y + z)/2 m; v = 10^{-3}·(x^{ }+ 2·y + z)/2 m; w = 10^{-3}·(x^{ }+ y + 2·z)/2 m.
Model description:
The system is modeled by 4 volumetric elements (FE type is 36) and by 6 volumetric elements (FE type is 34).
Analytical solution:
Calculation results:
Target value |
Analytical solution |
LIRA 10 |
Deviation, % |
σ_{x}, Pa |
2000000 |
2000000 |
0.00 |
σ_{y}, Pa |
2000000 |
2000000 |
0.00 |
σ_{z}, Pa |
2000000 |
2000000 |
0.00 |
τ_{xy}, Pa |
400000 |
400000 |
0.00 |
τ_{xz}, Pa |
400000 |
400000 |
0.00 |
τ_{yx}, Pa |
400000 |
400000 |
0.00 |