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.
E, M (2021). Finite Element Algorithms For The Static And Dynamic Analysis Of Time-Dependent And Time-Independent Plastic Bodies. Afribary. Retrieved from https://tracking.afribary.com/works/finite-element-algorithms-for-the-static-and-dynamic-analysis-of-time-dependent-and-time-independent-plastic-bodies
E, MODIFY "Finite Element Algorithms For The Static And Dynamic Analysis Of Time-Dependent And Time-Independent Plastic Bodies" Afribary. Afribary, 15 May. 2021, https://tracking.afribary.com/works/finite-element-algorithms-for-the-static-and-dynamic-analysis-of-time-dependent-and-time-independent-plastic-bodies. Accessed 09 Nov. 2024.
E, MODIFY . "Finite Element Algorithms For The Static And Dynamic Analysis Of Time-Dependent And Time-Independent Plastic Bodies". Afribary, Afribary, 15 May. 2021. Web. 09 Nov. 2024. < https://tracking.afribary.com/works/finite-element-algorithms-for-the-static-and-dynamic-analysis-of-time-dependent-and-time-independent-plastic-bodies >.
E, MODIFY . "Finite Element Algorithms For The Static And Dynamic Analysis Of Time-Dependent And Time-Independent Plastic Bodies" Afribary (2021). Accessed November 09, 2024. https://tracking.afribary.com/works/finite-element-algorithms-for-the-static-and-dynamic-analysis-of-time-dependent-and-time-independent-plastic-bodies