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Erschienen in:
2018 | OriginalPaper | Buchkapitel
8. Incompressible Plane Strain Elements: Locking-Free in the x and y Directions
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Abstract
A procedure, which takes the Poisson’s ratio ν to be exactly 1∕2, is developed here by selecting only those Rayleigh mode vectors that point-wise satisfy the isochoric, i.e., zero-volume-change, condition. A more research-type exposé can be found in Dasgupta (Acta Mech 223:1645–1656, 2012). A continuum mechanics treatment of the isochoric formulation in Cartesian coordinates can be found in Spencer (Continuum mechanics. Longman, Harlow, 1980) (Spencer indicated the dilatation by Δ; we use Θ instead, in this textbook.).
For four-node plane strain incompressible elements, the Rayleigh polynomial vectors, which satisfy equilibrium point-wise, are associated with:
1.
three rigid body modes, which trivially satisfy incompressibility
2.
uniform deviatoric stresses (2 modes), and these necessarily conform with the incompressibility condition
3.
two incompressible linearly varying axial strains without shear (ε
xx
+ ε
yy
= 0, ε
xy
= 0; ε
xx
, ε
yy
: linear combinations of (x, y).).
Within each element level, a uniform element pressure must be taken as the eighth independent variable. Interestingly, the pseudoinversion (in Sect. 1.8, the Moore–Penrose weak inverse of rectangular matrices is described) of the modal-to-nodal (eight rows by seven columns) rectangular matrix makes this chapter very distinct from popular isochoric formulations.
Incompressibility disqualifies the nodal displacements from being counted as the degrees-of-freedom. An independent nodal displacement invariably alters the element volume. Hence, the equilibrium and nodal compatibility equations are solved by determining the weights of the seven Rayleigh mode vectors and one (constant) pressure variable per element (To facilitate symbolic programming an additional notation to indicate the element number, which is encased within superscript parentheses, has been introduced.). Thus, without assembling the global stiffness matrix, all nodal displacements as well as all unknown element pressures are determined. As usual, concavity does not pose any difficulty.