Solid elements with rotational degrees of freedom: Part II—tetrahedron elements

Abstract

This is the second part of a two part paper on three‐dimensional finite elements with rotational degrees of freedom (DOF). Part II introduces a solid tetrahedron element having 3 translational and 3 rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 10‐node tetrahedron element. To further enhance the element performance a least squares strain extraction technique is also implemented to develop the stiffness matrix with a desired field. The strain smoothing improves performance without causing a loss in generality. As with the hexahedron in Part I, the element stiffness is augmented with a small penalty stiffness to eliminate any possible spurious zero energy modes. The new tetrahedron element passes the patch test and demonstrates much improved performance over the 4‐node translational DOF only (constant strain) tetrahedron element.

Hybrid stress tetrahedral elements with Allman's rotational D.O.F.s

“…They are termed zero-energy rotation modes. These modes have also been noted by Pawlak et al [8]. Despite of the fact that an e!ective scheme for stabilizing the four zero-energy rotation modes in the tetrahedral element has been developed [8], the fourth zero-energy rotation mode remains unidenti"ed.…”

“…TET4R*the displacement-based element [8]. TET4RX*the element proposed by Pawlak et al [8]; the element has an assumed strain "eld which is determined by minimizing the square of the di!erence between the assumed and displacement-derived strain "elds; the element is available in ANSYS as the SOLID72 element; it should be remarked that the element is not frame invariant [22] and exhibit zero-energy deformation modes, see the appendix.…”

“…TET4RX*the element proposed by Pawlak et al [8]; the element has an assumed strain "eld which is determined by minimizing the square of the di!erence between the assumed and displacement-derived strain "elds; the element is available in ANSYS as the SOLID72 element; it should be remarked that the element is not frame invariant [22] and exhibit zero-energy deformation modes, see the appendix. HT4R18*the hybrid stress element employing the 18 symmetric stress modes given in Equation (15); the zero-energy rotation modes are suppressed by the four skew symmetric stress modes given in Equation (24); the constant and higher order symmetric stress modes are orthogonal, see Section 6.…”

“…While a number of advanced hexahedral elements equipped with Allman's rotation have been available [7,9,17], there is only one advanced tetrahedral element model which is equipped with Allman's rotation [8], to the best knowledge of the authors. The model, known as TET4RX, has been available in the element library of ANSYS as its SOLID72 element.…”

A family of triangular and tetrahedral elements with rotational degrees of freedom

“…If desired, the elements could easily be stabilized by regularizing the element matrices. It should be noted that in [13,17] also tetrahedral elements with rotational degrees of freedom have been developed, but this derivation was based on mid-side nodes rather than nodal rotations only. …”