To Generalize Carath\'eodory's Continuity Theorem

Let $\varphi: D\rightarrow \Omega$ be a homeomorphism from a circle domain $D$ onto a domain $\Omega\subset\hat{\mathbb{C}}$. We obtain necessary and sufficient conditions (1) for $\varphi$ to have a continuous extension to the closure $\overline{D}$ and (2) for such an extension to be injective. Further assume that $\varphi$ is conformal and that $\partial\Omega$ has at most countably many non-degenerate components $\{P_n\}$ whose diameters have a finite sum $\displaystyle\sum_n{\rm diam}(P_n)<\infty$. When the point components of $\partial D$ or those of $\partial \Omega$ form a set of $\sigma$-finite linear measure, we can show that $\varphi$ continuously extends to $\overline{D}$ if and only if all the components of $\partial\Omega$ are locally connected. This generalizes Carath\'eodory's Continuity Theorem, that concerns the case when $D$ is the open unit disk $\left\{z\in\hat{\mathbb{C}}: |z|<1\right\}$, and allows us to derive a new generalization of the Osgood-Taylor-Caratheodry Theorem. Here a Peano compactum is a compact metrisable space with locally connected components such that for any $C>0$ at most finitely many of its components are of diameter greater than $C$.Comment: 39 pages, 3 figure

The fatigue properties and damage of the corroded steel bars under the constant-amplitude fatigue load

We obtained the corroded steel bars by conducting electrically-accelerated corrosion tests. Then, to investigate the effects of the corrosion ratio and the stress amplitude on the fatigue life, and to further study the damage evolution law under corrosion and fatigue loads, we performed axial fatigue tests on 13 steel bars with various corrosion ratios. The laboratory results show that the fatigue life is logarithmical linear to the stress amplitude, and the increase in corrosion ratio leads to the accelerated decrease in the fatigue life. In addition, the increase in stress amplitude can accelerate the fatigue damage, and further decreases the fatigue life. With the laboratory data, we further established a model to predict the fatigue life of the steel bars with various corrosion ratios. The evolution of the residual strains includes the relatively rapid increase, stable increase and rapid increase stages. Moreover, we developed an evolution equation for the residual strain, and this equation can properly describe the laboratory results. Furthermore, considering the fatigue damage, we proposed a constitutive model to describe the stress-strain curve of the corroded steel bar under static tension. The comparison shows that the calculated stress curves agree well with the laboratory curves