April 04, 2008
A brief history
The earliest paper on non-linear finite elements appears to be that by Turner et ul. which dates from 1960 and, significantly, stems from the aircraft industry. The present review will cover material published within the next twelve years (up to and including 1972).
Most of the other early work on geometric non-linearity related primarily to the linear buckling problem and was undertaken by amongst others. For genuine geometric non-linearity, ‘incremental’ procedures were originally adopted using the ‘geometric stiffness matrix’ in conjunction with an updating of coordinates and, possibly, an initial displacement matrix. A similar approach was adopted with material non-linearity. In particular, for plasticity, the structural tangent stiffness matrix (relating increment of load to increments of displacement) incorporated a tangential modular matrix which related the increments of
stress to the increments of strain.
Unfortunately, the incremental (or forward-Euler) approach can lead to an unquantifiable build-up of error and, to counter this problem, Newton-Raphson iteration was used by, amongst others, Mallet and Marcal and Oden. Direct energy search methods were also adopted. A modified Newton-Raphson procedure was also recommended by Oden, Haisler and Zienkiewicz. In contrast to the full Newton-Raphson method, the stiffness matrix would not be continuously updated. A special form using the very initial, elastic stiffness matrix was referred to as the ‘initial stress’ method and much used with material non-linearity. Acceleration procedures were also considered. The concept of combining incremental (predictor) and iterative (corrector) methods was introduced by Brebbia and Connor and Murray and Wilson who thereby adopted a form of ‘continuation method’.
Early work on non-linear material analysis of plates and shells used simplified methods with sudden plastification. Armen traced the elasto-plastic interface while layered or numerically integrated procedures were adopted by, amongst others, Marcal and Whang combined material and geometric non-linearity for plates initially involved ‘perfect elasto-plastic buckling’. One of the earliest fully combinations employed an approximate approach and was due to Murray and Wilson. A more rigorous ‘layered approach’ was applied to plates and shells by Marcal, Gerdeen and Striklin. Various procedures were used for integrating through the
depth from a ‘centroidal approach’ with fixed thickness layers to trapezoidal and Simpson’s rule. To increase accuracy, ‘sub-increments’ were introduced for plasticity by Nayak and Zienkiewicz. Early work involving ‘limit points‘ and ‘snap-through’ was due to Sharifi and Popov and Sabir and Lock.
Copyright: M.A. Crisfield - Non-linear Finite Element analysis of Solids and Structures.
