D.R.J. Owen

Design of simple low order finite elements for large strain analysis of nearly incompressible solids

A simple four-node quadrilateral and an eight-node hexahedron for large strain analysis of nearly incompressible solids are proposed. Based on the concept of deviatoric/volumetric split and the replacement of the compatible deformation gradient with an assumed modified counterpart, the formulation developed is applicable to arbitrary material models. The closed form of the corresponding exact tangent stiffnesses, which have a particularly simple structure, is derived. It ensures asymptotically quadratic rates of convergence of the Newton-Raphson scheme employed in the solution of the implicit finite element equilibrium equations. From a practical point of view, the incorporation of the proposed elements into existing codes is straightforward. It requires only small changes in the routines of the standard displacement based 4-node quadrilateral and 8-node brick. A comprehensive set of numerical examples, involving hyperelasticity as well as multiplicative elasto-plasticity, is provided. It illustrates the performance of the proposed elements over a wide range of applications, including strain localisation problems, metal forming simulation and adaptive analysis.

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.

Transfer operators for evolving meshes in small strain elasto-placticity

Abstract Together with error evaluation and mesh refinement, transfer operators represent the essential component of the adaptive procedure applied to history-dependent materials. Aspects of the transfer operation discussed in this work relate to: equilibrium conditions, consistency with the constitutive equation, geometric issues and diffusion of state variables between successive meshes. In this paper attention is restricted to small strain rate-independent elasto-plasticity. Numerical examples are presented to illustrate some practical features of the computational procedure.

A refined analysis of laminated plates by finite element displacement methods. I: Fundamentals and static analysis

Abstract This and a companion paper ( Computers and Structures 26 , 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.

Infinite elements in quasi-static materially nonlinear problems

Abstract Unbounded continua arise in many engineering problems, particularly those involving geo-mechanical situations. The traditional engineering approach of simple truncation introduces problems of positioning the finite boundary for accurate solution. In this paper mapped infinite elements are coupled to conventional elements in order to model the far field response and attention is focussed on problems in which plastic flow takes place. Various families of mapped infinite elements are considered and details of their formulation and computer implementation are provided. The applicability of this finite/infinite element approach is assessed by the solution of both two and three dimensional elasto-plastic examples.

A combined finite/discrete element simulation of shot peening processes – Part II: 3D interaction laws

Following earlier work on the combined finite/discrete element simulation of shot peening process in 2D case, 3D representation of the problem is established with respect to DE modelling and contact interaction laws. An important relevant computational issue regarding the critical time step is carefully studied, and a new time stepping scheme that can ensure both short and long term stability of the contact models is developed. Numerical tests are performed to evaluate the proposed normal and frictional contact interaction laws with various model parameters. The influences of single and multiple shot impact, as well as element sizes are also numerically investigated. The established contact interaction laws can also be applied to other multi‐body dynamic simulations.

A combined finite/discrete element simulation of shot peening processes – Part I: studies on 2D interaction laws

In the first part of this series of papers on the combined finite/discrete element simulation of shot peening processes, different contact interaction laws for 2D cases are extensively studied with special attention given to the proper selection of the parameter values involved, which is one of the key issues for successful direct simulation. In addition, computational issues including contact forces, partial contact, energy dissipation, and rheological representation are addressed. Numerical examples for a single shot impact system simulated by the coupled finite/discrete element method using different interaction laws are provided to verify the proposed approaches. The results are also compared with those obtained by using only finite element methods. Findings obtained by performing 2D simulations will, in the subsequent article, be used in realistic computational simulations of 3D shot peening processes.

A refined analysis of laminated plates by finite element displacement methods—II. Vibration and stability

Abstract A refined anisotropic laminated plate bending analysis is presented in a companion paper to the present work ( Computers and Structures 26 , 907–914, 1987). In the present paper, a refined transverse vibration and buckling analysis based on the same local model is presented. The numerical examples presented are compared with analytical solutions and classical plate theory and it is demonstrated that the present model predicts a realistic laminate global response.

Discrete thermal element modelling of heat conduction in particle systems: Basic formulations

This paper proposes a novel numerical methodology, termed the discrete thermal element method (DTEM), for the effective modelling of heat conduction in systems comprising a large number of circular particles in 2D cases. Based on an existing analytical integral solution for the temperature distribution over a circular domain subjected to the Neumann boundary condition, a linear algebraic system of thermal conductivity equations for each particle is derived in terms of the average temperatures and the resultant fluxes at the contact zones with its neighboring particles. Thus, each particle is treated as an individual element with the number of (temperature) unknowns equal to the number of particles that it is in contact with. The element thermal conductivity matrix can be very effectively evaluated and is entirely dependent on the characteristics of the contact zones, including the contact positions and contact angles. This new element shares the same form and properties with its conventional thermal finite element counterpart. In particular, the whole solution procedure can follow exactly the same steps as those involved in the finite element analysis. Unlike finite elements or other modern numerical techniques, however, no discretization errors are involved in the discrete thermal elements. The modelling error mainly stems from the assumption made about the heat flux distribution within the contact zones. Based on some theoretical work, an enhanced version is suggested to improve the approximation. The numerical assessment against finite element results indicates that the basic version of DTEM can achieve a very reasonable solution accuracy, while the enhanced version further improves the accuracy to a high level. In addition, thermal resistance phenomena between the contact zones can be readily incorporated into the current modelling framework.

Finite element analysis of reinforced and prestressed concrete structures including thermal loading

Abstract This paper describes the application of finite element techniques to the solution of nonlinear concrete problems. Reinforced concrete thick plates and shells are first considered for which both a perfect and strain-hardening plasticity approach are employed to model the compressive behaviour. A dual criterion for yielding and crushing in terms of stresses and strains is considered, which is complemented with a tension cut-off representation. Degenerate thick shell elements employing a layered discretisation through the thickness are adopted and both reduced and selectively integrated 8-node serendipity and heterosis elements are considered. Thermal loading of prestressed concrete structures is also considered which necessitates the inclusion of time effects in the analysis. The technique described in this paper involves concurrently solving an uncoupled set of equations within a time interval to provide both the displacement and temperature increments. A two-level time stepping scheme is employed to predict temperature changes within a time interval and elasto-viscoplastic material analysis is performed using an explicit forward-difference scheme incorporating an equilibrium iteration procedure. The constitutive model for the concrete is essentially identical to that employed for the plate and shell analysis. Numerical examples are presented for both types of analysis and comparison is made with experimental results whenever possible. Additionally, results for thermal loading are presented which indicate that a full transient thermal-mechanical analysis is sometimes essential in order to obtain a realistic structural response.