Notch sensitivity of circular rings subjected to impact loading

'Notch-sensitive regions' have been observed during a series of experimental investigations into the dynamic plastic behaviour and failure of thin-walled metallic radially notched circular rings with are-shaped supports subjected to concentrated impact loads. The experimental results show that the exterior notches at some regions have no effect on the deformation of the rings, but do have effect at the remaining regions. The notch-sensitive region is theoretically determined by using the equivalent structures technique; fairly good agreement has been reached between the simple theory and the experimental results. Both dimensional and theoretical analyses prove that whether a plastic hinge formed or not at the notched section does not depend on the mean radius of the ring and the input kinetic energy. It depends on the weak coefficient of the notched section and the angle of the support. Generally speaking, there are mainly three failure modes for a notched circular ring with are-shaped support under impact loading: Mode I, large inelastic deformation when the notch is outside the sensitive region, in this case the ring deforms as a normal one; Mode II, large inelastic deformation only at some part of the ring and tearing occurred at the notched sections; Mode III, large inelastic deformation and total rupture occurred at the notched sections. It is believed that the present study could assist the understanding of the dynamic behaviour and failure of other kinds of nonstraight components with macroscopic imperfections under impulsive loading.

On the invariant representation of spin tensors with applications

The invariant representation of the spin tensor defined as the rotation rate of a principal triad for a symmetric and non-degenerate tensor is derived on the basis of the general solution of a linear tensorial equation. The result can be naturally specified to study the. spin of the stretch tensors and to investigate the relations between various rotation rate tensors encountered frequently in modern continuum mechanics. A remarkable formula which relates the generalized stress conjugate to the generalized strain in Hill's sense. to Cauchy stress, is obtained in invariant form through the work conjugate principle. Particularly, a detailed discussion on the time rate of logarithmic strain and its conjugate stress is made as the principal axes of strain arc not fixed during deformation.

Constant mean curvature hypersurfaces with constant δ-invariant

We completely classify constant mean curvature hypersurfaces (CMC) with constant δ-invariant in the unit 4-sphere S4 and in the Euclidean 4-space E4.

Invariant-free deduction systems for temporal logic

In this thesis we propose a new approach to deduction methods for temporal logic. Our proposal is based on an inductive definition of eventualities that is different from the usual one. On the basis of this non-customary inductive definition for eventualities, we first provide dual systems of tableaux and sequents for Propositional Linear-time Temporal Logic (PLTL). Then, we adapt the deductive approach introduced by means of these dual tableau and sequent systems to the resolution framework and we present a clausal temporal resolution method for PLTL. Finally, we make use of this new clausal temporal resolution method for establishing logical foundations for declarative temporal logic programming languages. The key element in the deduction systems for temporal logic is to deal with eventualities and hidden invariants that may prevent the fulfillment of eventualities. Different ways of addressing this issue can be found in the works on deduction systems for temporal logic. Traditional tableau systems for temporal logic generate an auxiliary graph in a first pass.Then, in a second pass, unsatisfiable nodes are pruned. In particular, the second pass must check whether the eventualities are fulfilled. The one-pass tableau calculus introduced by S. Schwendimann requires an additional handling of information in order to detect cyclic branches that contain unfulfilled eventualities. Regarding traditional sequent calculi for temporal logic, the issue of eventualities and hidden invariants is tackled by making use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automation. A remarkable consequence of using either a two-pass approach based on auxiliary graphs or aone-pass approach that requires an additional handling of information in the tableau framework, and either invariant-based rules or infinitary rules in the sequent framework, is that temporal logic fails to carry out the classical correspondence between tableaux and sequents. In this thesis, we first provide a one-pass tableau method TTM that instead of a graph obtains a cyclic tree to decide whether a set of PLTL-formulas is satisfiable. In TTM tableaux are classical-like. For unsatisfiable sets of formulas, TTM produces tableaux whose leaves contain a formula and its negation. In the case of satisfiable sets of formulas, TTM builds tableaux where each fully expanded open branch characterizes a collection of models for the set of formulas in the root. The tableau method TTM is complete and yields a decision procedure for PLTL. This tableau method is directly associated to a one-sided sequent calculus called TTC. Since TTM is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus TTC is also free from this kind of structural rules. In particular, TTC is free of any kind of cut, including invariant-based cut. From the deduction system TTC, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PLTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to temporal logic. The most fruitful approach in the literature on resolution methods for temporal logic, which was started with the seminal paper of M. Fisher, deals with PLTL and requires to generate invariants for performing resolution on eventualities. In this thesis, we present a new approach to resolution for PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Our method is based on the dual methods of tableaux and sequents for PLTL mentioned above. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called TRS-resolution, that extends classical propositional resolution. Finally, we prove that TRS-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL. In the field of temporal logic programming, the declarative proposals that provide a completeness result do not allow eventualities, whereas the proposals that follow the imperative future approach either restrict the use of eventualities or deal with them by calculating an upper bound based on the small model property for PLTL. In the latter, when the length of a derivation reaches the upper bound, the derivation is given up and backtracking is used to try another possible derivation. In this thesis we present a declarative propositional temporal logic programming language, called TeDiLog, that is a combination of the temporal and disjunctive paradigms in Logic Programming. We establish the logical foundations of our proposal by formally defining operational and logical semantics for TeDiLog and by proving their equivalence. Since TeDiLog is, syntactically, a sublanguage of PLTL, the logical semantics of TeDiLog is supported by PLTL logical consequence. The operational semantics of TeDiLog is based on TRS-resolution. TeDiLog allows both eventualities and always-formulas to occur in clause heads and also in clause bodies. To the best of our knowledge, TeDiLog is the first declarative temporal logic programming language that achieves this high degree of expressiveness. Since the tableau method presented in this thesis is able to detect that the fulfillment of an eventuality is prevented by a hidden invariant without checking for it by means of an extra process, since our finitary sequent calculi do not include invariant-based rules and since our resolution method dispenses with invariant generation, we say that our deduction methods are invariant-free.