Peirce algebras

We present a two-sorted algebra, called aPeirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a set-forming operator on relations (the Peirce product of Boolean modules) and a relation-forming operator on sets (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-calledterminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

逐步求精的一种模型

提出一种支持程序开发的结构化方法．该方法以一种简单的问题分解策略为基础，比Ｗｉｒｔｈ－Ｄｉｊｋｓｔｒａ的自顶向下逐步求精方法更利于面向目标的程序设计．由该方法可知，一个程序可经一系列求精而开发出来，每一步求精都能为相应的最弱前置条件序列建立后置条件．这种策略使情况分析减少到极限，简化了结构化程序的证明，并保证了程序结构和数据结构之间的对应．     

The toughness of laser welded joints in the ductile-brittle transition

An important feature of laser welded joints is the high level of strength and toughness mismatch developed between the weld and the parent plate. As a result, defects are frequently located in regions with high strength and toughness gradients. To address this issue, toughness tests have been performed on laser welded joints with strength mismatches in excess of 2. Toughness tests have been performed on high and low constraint mode I geometries and mixed-mode I/II configurations. In highly constrained geometries, the local toughness dominates the failure process causing the crack to deviate into harder and more brittle weld metal, while in low constraint configurations, the size of the plastic zone promotes crack deviation into the softer and tougher parent plate. In Charpy tests the crack also deviates into the tougher parent plate giving potentially misleading indications of the behaviour of defects in highly constrained joints. The experiments are modelled by a local approach technique for functionally graded materials in which the local yield strength and toughness are allowed to vary spatially.     

统计断裂理论的“应力独立作用”假设及几种修正形式

本文首先对统计分布函数进行简化处理,并给出了在多轴应力状态下的数值计算结果。在此基础上建立了“应力独立作用”假设的成立条件,讨论了它的使用范围及相应的误差,然后导出了使用更为广泛的几种修正形式,兼而讨论了按“方向平均应力”假设的正确性问题。     

ENGINEERING ESTIMATION OF THE LOWER BOUND TOUGHNESS IN THE TRANSITION REGIME OF FERRITIC STEELS

A simple engineering method for estimating a lower bound fracture toughness in the ductile-to-brittle transition region is presented. It is based on empirical evidence that the lower tail of the failure probability versus fracture toughness curve tends to be a straight line rather than a Weibull curve. Fitting the lower tail by a straight line, however, gives an engineering lower bound at a theoretical failure probability of zero. The method enables one to obtain lower bounds using small data sets of three or four specimens.     

Sample size dependence of flaw distributions for the prediction of brittle solids strength using additive Weibull bimodal distributions

A method to evaluate the mean strength of brittle solids at short gauge length from experiments performed at higher gauge length is proposed for bimodal fracture behavior. The method based on additive Weibull bimodal distributions takes the evolution of the relative proportion of flaws along the gauge length into account via a gauge length dependent mixing parameter. A linear dependence of this parameter vs. the gauge length is proposed. The approach is assessed using experimental results on carbon and E-glass fibers. The new method provides values for the average tensile strength of the fibers at 100 μm up to 27% higher than those calculated using the classical approach. The underestimation of the classical approach can be attributed to the weight of the severe category of flaws at short length that is considered to be the same as that determined at the experimental gauge length, which can be several orders higher. A simplified approach taking into account solely the more severe category of flaw is shown to be applicable for the prediction of the strength at short gauge length independently of the nature of the fiber.     

Safety and progress of recursive procedures

Temporal weakest precondions are introduced for calculational reasoning about the states encountered during execution of not-necessarily terminating recursive procedures. The formalism can distinguish error from useful nontermination. The precondition functions are constructed in a new and more elegant way. Healthiness laws are discussed briefly. Proof rules are introduced that enable calculational proofs of various safety and progress properties. The construction of the precondition functions is justified in an Appendix that provides the operational semantics.Dedicated to the memory of Jan van de Snepscheut     

Improved functional perturbation method for reliability analysis of randomly heterogeneous beams

The strength reliability of randomly heterogeneous beams is studied. The beams are considered as brittle, and failure by the weakest link criterion is assumed. The structure is statically indeterminate, thus the stress field is a function of the random morphology. The probabilistic beam strength is a coupled functional of morphology and stresses. Correlation between local strength and local modulus is also considered, and its effect on reliability is investigated. Heterogeneity is confined to the longitudinal direction only. An improved analytical solution is found by a new, optimized functional perturbation method (FPM). The improvement is achieved by two operations: generalizing the previously used FPM to account for multifunctional dependency, and choosing the perturbation to be around the proper homogeneous case for each type of morphology. It is shown that the improvement of the optimized method for reliability analysis is significant, and depends on the type of local strength-modulus statistical correlation. In addition, analytical results for very large and very small correlation lengths are obtained, and their validity range is examined by numerical solutions.     

ZRC – A Refinement Calculus for Z

The fact that Z is a specification language only, with no associated program development method, is a widely recognised problem. As an answer to that, we present ZRC, a refinement calculus based on Morgan's work that incorporates the Z notation and follows its style and conventions. This work builds upon existing refinement techniques for Z, but distinguishes itself mainly in that ZRC is completely formalised. In this paper, we explain how programs can be derived from Z specifications using ZRC. We present ZRC-L, the language of our calculus, and its conversion laws, which are concerned with the transformation of Z schemas into programs of this language. Moreover, we present the weakest precondition semantics of ZRC-L, which is the basis for the derivation of the laws of ZRC. More than a refinement calculus, ZRC is a theory of refinement for Z. Received July 1997 / Accepted in revised form October 1998