Finite Element Algorithms For The Static And Dynamic Analysis Of Time-Dependent And Time-Independent Plastic Bodies

ABSTRACT

Continuum and finite element formulations ofthe static and dynamic initial-boundaryvalue

evolution (elastoplastic) problems are considered in terms of both the classical

and internal variable frameworks. The latter framework is employed to develop

algorithms in the form of convex mathematical programming and Newton-Raphson

schemes. This latter scheme is shown to be linked to the former in the sense that it

expresses the conditions under which the convex non-linear function can be minimised.

A Taylor series expansion in time and space is extensively employed to derive

integration schemes which include the generalised trapezoidal rule and a generalised

Newton-Raphson scheme. This approach provides theoretical foundations for the

generalised trapezoidal rule and the generalised Newton-Raphson scheme that have

some geometrical insights as well as an interpretation in terms of finite differences and

calculus. Conventionally, one way of interpreting the generalised trapezoidal rule is

that it uses a weighted average of values (such as velocity or acceleration) at the two

ends ofthe time interval.

In this dissertation, the generalised trapezoidal scheme is shown to be a special case of

the forward-backward difference scheme for solving first order differential equations.

It includes the Euler forward and backward difference schemes as special cases when

the integration parameters are set to a = 0 and a = 1, respectively. For the solution of

second order differential equations, two schemes are developed, termed the FB] and

FB2 schemes. The generalised Newton-Raphson scheme includes the conventional

Newton-Raphson scheme as a special case when its integration scalars are set to f3 = 0

for an implicit version or f3 = 1 for an explicit version.

To consolidate the parametric studies of the integration schemes developed, stability

analyses are performed using the energy and spectral stability methods. Both methods

reveal that the FB] and FB2 schemes yield conditionally and unconditionally stable

algorithms depending on the choices ofintegration parameters.

Finite element numerical examples in the form of plane stress, plane strain and

axisymmetric models are used to evaluate the performance of the algorithms; which

demonstrate that the generalised Newton-Raphson scheme can considerably enhance

convergence ofthe conventional Newton-Raphson scheme by converging quadratically

even when a large time-step, (violating incremental strain laws) is used, without loss of

accuracy, thus providing considerable advantages over the conventional scheme by

reducing the costs of computations associated with an incremental process.