Title:

A review and the development of bounding methods in continuum mechanics

Energy theorems and kindred inequalities have long been a basis for the analysis of redundant structures and the material continuum. In the first section of this thesis we trace the development of the principal results of elasticity, timeindependent inelasticity and creep, from the principle of virtual work and the wellknown theorems of linear elasticity to recent results which describe the deformation of general inelastic materials under timevarying loads. In certain instances where incompleteness is apparent in the theory an attempt is made to remedy this; in particular we present a new view of the upper bound shakedown theorem  an area which remains relatively unexplored in comparison with the lower bound theorem and the limit theorems. A discussion of the fundamental material requirements which permit the establishment of many of the inequalities is included. In the following section we obtain new bounding results for a class of constitutive relations using a thermodynamic formalism as the basis of the discussion. The bounds turn out to be both simple in form and insensitive to the detailed aspects of the material behaviour. Cyclic work bounds are derived in which the cyclic stress history known as the "rapid cycle solution" gives a simple physical meaning to the bounding results. Examples are given for linear viscoelastic models, the nonlinear viscous model and the BaileyOrowan recovery model. A displacement bound is derived which is expressed in terms of two plasticity solutions and the result of a simple creep test. Examples are given and the results we obtain for the Bree problem are compared with O'Donnell's solutions which are in use in current design. In the third section, new results are obtained for the behaviour of a general viscoelastic material subjected to cyclic loading. The existence and uniqueness of a stationary cyclic state of stress is proved and a lower work bound for the general nonlinear material is derived. An upper work bound is obtained for the general linear material in terms of the rapid cycle solution and we describe a simple method for obtaining this solution without the need for a full analysis. The role of the constitutive equation in the bounding theory is investigated when the method based on a state variable description is compared with the results obtained from the use of a historydependent constitutive relation. We go on to show how a knowledge of the response of a viscoelastic body to constant loading is sufficient to determine its general longterm cyclic strain behaviour. In the final section we bring together the existing theorems concerning small deformations of timedependent materials and large deformations of timeindependent materials. The problem posed has dual complexity as a result of the dependence of the deformation on the stress history and the dependence of the stress on the changing geometry. We obtain a general displacement bound in terms of suitably defined conjugate variables referred to the undeformed configuration. In an example which follows it is shown that the employment of such variables may in some cases reduce the difficulty of bounding nonlinear deformations to a level that is comparable with the linear case.
