Samuel W. Key

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

A family of uniform strain elements is presented for three-node triangular and four-node tetrahedral meshes. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favorable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three-node triangular or four-node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behavior for a set of example problems.

A variational principle for incompressible and nearly-incompressible anisotropic elasticity

Abstract A specialized form of Reissners variational principle is developed which is suitable for anisotropic incompressible and nearly-incompressible thermoelasticity. The finite element method is used to find solutions in two axisymmetric problems where the material is cylindrically orthotropic and incompressible. The developed variational principle has the feature that the volumetric strain appears in only one equation and errors committed in approximating this equation do not re-enter the stress calculations.

Methods for connecting dissimilar three-dimensional finite element meshes†

Two methods are presented for connecting dissimilar three-dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first-order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non-planar surfaces. Both methods can be used to connect meshes with different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three-dimensional linear elasticity are presented. Published in 2000 by John Wiley & Sons, Ltd.

On the numerical implementation of inelastic time dependent and time independent, finite strain constitutive equations in structural mechanics☆

Abstract A number of complex issues are resolved in a way that allows the incorporation of finite strain, inelastic material behavior into the piecewise numerical construction of solutions in solid mechanics. Without recourse to extensive continuum mechanics preliminaries, an elementary time independent plasticity model, an elementary time dependent creep model, and a viscoelastic model are introduced as examples of constitutive equations which are routinely used in engineering calculations. The constitutive equations are all suitable for problems involving large deformations and finite strains. The plasticity and creep models are in rate form and use the symmetric part of the velocity gradient or the stretching to compute the co-rotational time derivative of the Cauchy stress. The viscoelastic model computes the current value of the Cauchy stress from a hereditary integral of a ‘materially invariant’ form of the stretching history. The current configuration is selected for evaluation of equilibrium as opposed to either the reference configuration or the last established equilibrium configuration. The process of strain incrementation is examined in some depth. The stretching, evaluated at the mid-interval and multiplied by the time step, is identified as the appropriate finite strain increment to use with the selected forms of the constitutive equations. Discussed is the conversion of rotation rates based on the spin into incremental orthogonal rotations. These rotations are used to update stresses and state variables due to rigid body rotation during the load increment. Comments and references to the literature are directed at numerical integration of the constitutive equations with an emphasis on doing this accurately, if not exactly, for any time step and stretching. This material taken collectively provides an approach to numerical implementation which is marked by its simplicity.

A finite element procedure for the large deformation dynamic response of axisymmetric solids

Abstract A computational procedure for the large deformation dynamic response of solids is presented. The underlying mechanics, the constitutive theories of interest, the spatial discretization, and the time integration scheme are each discussed. The mechanics is carried out in the current configuration described by a fixed spatial coordinate system and using the Cauchy stress. Elastic, elastoplastic, viscoelastic and curshable foam constitutive theories are examined. A bilinear isoparametric quadrilateral finite element is employed for the spatial discretization. An explicit central difference time integration scheme and artificial viscosity are used to compute the response. The results of five computations are presented and compared with either experimental results or exact answers.

A method for connecting dissimilar finite element meshes in two dimensions

A method is presented for connecting dissimilar finite element meshes in two dimensions. The method combines standard master–slave concepts with the uniform strain approach for finite elements. By modifying the definition of the slave boundary, corrections are made to element formulations such that first-order patch tests are passed. The method can be used to connect meshes which use different element types. In addition, master and slave boundaries can be designated independently of relative mesh resolutions. Example problems in two-dimensional linear elasticity are presented. Copyright

An improved constant membrane and bending stress shell element for explicit transient dynamics

Abstract This paper presents an improved constant stress shell element for explicit transient analysis. The new shell element is based on the popular Belytschko-Lin-Tsay shell which is the most commonly used shell element in explicit transient analysis today because of its superior efficiency. Although the Belytschko-Lin-Tsay shell is well known for its accuracy in the class of mean quadrature elements, it exhibits performance deficiencies when the initial geometry of the element is warped and when the element is subjected to warping distortions. The new shell presented here provides improved accuracy by considering warped geometry with discrete normals at the element nodal points. Physical stiffness in the warping mode is introduced to improve the element behavior for warping distortions. The improvements are achieved with only a moderate increase in computational cost, while still preserving the membrane and bending behavior of the Belytschko-Lin-Tsay shell.

A transition element for uniform strain hexahedral and tetrahedral finite elements

A transition element is presented for meshes containing uniform strain hexahedral and tetrahedral finite elements. It is shown that the volume of the standard uniform strain hexahedron is identical to that of a polyhedron with 14 vertices and 24 triangular faces. Based on this equivalence, a transition element is developed as a simple modification of the uniform strain hexahedron. The transition element makes use of a general method for hourglass control and satisfies first-order patch tests. Example problems in linear elasticity are included to demonstrate the application of the element. Copyright

Dynamic Relaxation Applied to the Quasi-Static, Large Deformation, Inelastic Response of Axisymmetric Solids

The use of dynamic relaxation as a solution strategy for the quasi-static, large deformation, inelastic response of solids is examined. The underlying mechanics, the constitutive theories of interest, the incremental form of the equations, the spatial discretization, and the implementation of dynamic relaxation for path and/or time dependent material response are each discussed. The mechanics are carried out in the current configuration of the body described by a fixed spatial coordinate system and using the Cauchy stress. Finite strain constitutive theories for elastic, elastoplastic, and creep behavior are introduced. An incremental form of the problem allowing a sequence of equilibrium solutions to be found is presented. A constant bulk strain, bilinear displacement isoparametric quadrilateral finite element is employed for the spatial discretization. The solution strategy used to generate the sequence of equilibrium solutions is dynamic relaxation which in the form adopted is based on explicit central difference pseudo-time integration and artificial damping. It is used to find the next solution as a result of an increment in time and/or load. Each solution must satisfy equilibrium to within a prescribed tolerance before proceeding to the next increment. Several example calculations are presented.

A least-squares approach for uniform strain triangular and tetrahedral finite elements†

A least-squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements. The basis for the method is a weighted least-squares formulation in which a linear displacement field is fit to an elements nodal displacements. By including a greater number of nodes on the element boundary than is required to define the linear displacement field, it is possible to eliminate volumetric locking common to fully integrated lower-order elements. Such results can also be obtained using selective or reduced integration schemes, but the present approach is fundamentally different from those. The method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex, mid-edge and mid-face nodes. Example problems in two- and three-dimensional linear elasticity are presented. Element types considered in the examples include a six-node triangle, eight-node tetrahedron, and ten-node tetrahedron.