On stability of elastic domain during isothermal solid-solid phase transformationin a tube configuration

Under isothermal quasi-static stretching the phase transition of a superelastic NiTi tube involves the formation （during loading） and vanishing （in unloading） of a high strain （martensite） domain. The two events are accompanied by a rapid stress drop/rise due to the formation/vanishing of do- main fronts. From a thermodynamic point of view, both are instability phenomena that occur once the system reaches its critical state. This paper investigates the stability of a shrink- ing cylindrical domain in a tube configuration during unload- ing. The energetics and thermodynamic driving force of the cylindrical domain are quantified by using an elastic inclu- sion model. It is demonstrated that the two domain fronts ex- hibit strong interaction when they come close to each other, which brings a peak in the total energy and a sign change in the thermodynamic driving force. It is proved that such domain front interaction plays an important role in control- ling the stability of the domain and in the occurrence of stress jumps during domain vanishing. It is also shown that the pro- cess is governed by two nondimensional length scales （the normalized tube length and normalized wall-thickness） and that the length scale dependence of the critical domain length and stress jump for the domain vanishing can be quantified by the elastic inclusion model.

BOUNDARY INTEGRAL FORMULAS FOR ELASTIC PLANE PROBLEM OF EXTERIOR CIRCULAR DOMAIN

After the stress function and the normal derivative on the boundary for the plane problem of exterior circular domain are expanded into Laurent series, comparing them with the Laurent series of the complex stress function and making use of some formulas in Fourier series and the convolutions, the boundary integral formula of the stress function is derived further. Then the stress function can be obtained directly by the integration of the stress function and its normal derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function is convenient to be used for solving the elastic plane problem of exterior circular domain.

STABILITY OF DISPLACEMENT TO THE SECOND FUNDAMENTAL PROBLEM IN PLANE ELASTICITY

In this article, by using the stability of Cauchy type integral when the smooth perturbation for integral curve and the Sobolev type perturbation for kernel density happen, we discuss the stability of the second fundamental problem in plane elasticity when the smooth perturbation for the boundary of the elastic domain （unit disk） and the Sobolev type perturbation for the displacement happen. And the error estimate of the displacement between the second fundamental problem and its perturbed problem is obtained.

A STOCHASTIC ALGORITHM FOR FAULT INVERSE PROBLEMS IN ELASTIC HALF SPACE WITH PROOF OF CONVERGENCE

A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in[11].We show in this paper how it can be used to solve the fault inverse problem,where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements.With the parameter giving the plane containing the fault denoted by m and the regularization parameter for the linear part of the inverse problem denoted by C,both modeled as random variables,we derive a formula for the posterior marginal of m.Modeling C as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value[11].We prove that this posterior marginal of m is convergent as the number of measurement points and the dimension of the space for discretizing slips increase.Simply put,our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense.Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded.We also explain how our proof can be extended to a whole class of inverse problems,as long as some basic requirements are met.Finally,we show numerical simulations that illustrate the numerical convergence of our algorithm.