Solution of The Double-Coupled Problem of Buckling of a Shape Memory Alloy Rod Due to The Direct Thermoelastic Phase Transformation

Problems of the stability of a shape memory alloy rod during the direct martensitic phase transformation under constant compressive loading are solved taking into account the effect of stresses on the phase transition and the effect of the phase transition on the temperature regime using various statements. A comparison of the results of the obtained solutions is made.     

Two-Way Coupled Statement of the Problem of Loss of Stability due to Inverse Thermoelastic Phase Transition in a Shape Memory Alloy

A two-way coupled statement of stability problem for shape memory alloy elements is given in the framework of the “fixed load” and “variable load” concepts. It is shown that the largest values of the critical parameters are obtained when solving the problem in the two-way coupled statement in the framework of the “fixed load” concept and the least values are obtained in the oneway coupled statement in the framework of the “variable load” concept.     

形状记忆合金薄板的分叉与激变

研究了受横向载荷作用的形状记忆合金矩形薄板的非线性动力学特性及混沌行为.基于形状记忆合金材料的热-机耦合行为和拟弹性行为的五次多项式本构关系及薄板的动力学平衡方程,建立了反映形状记忆合金薄板动力学行为的非线性动力学模型;用平衡态定性分析法讨论了矩形薄板的动力学稳定性与材料相转换间的关系.利用数值模拟的方法,研究了形状记忆合金薄板动力系统的分叉和激变.分叉分析研究了随外加载荷变化和温度变化这两种情况时,系统的分叉结构.通过所绘制的分叉图和局部区域放大的分叉图,发现系统呈现出倍周期分叉、倒置的倍周期分叉、激变和突减等分叉现象.     

Solution of a Spatial Stress Problem for an Elastic Layer

The general solution to the spatial stress problem for an elastic layer is given in a new form. A layer compressed by normal forces uniformly distributed over a square is considered as an example.     

Phase-Transformation Fronts Evolution for Stress- and Strain-Controlled Tension Tests in TiNi Shape Memory Alloy

Nucleation and development of phase transformation fronts in TiNi shape memory alloy subjected to the stress- and strain-controlled tension tests were investigated. A thermovision camera was applied to register the distribution of infrared radiation emitted by the specimen and to find its temperature variations. During the loading, narrow bands of considerably higher temperature corresponding to the martensitic phase, starting from the central part of the specimen and developing towards the specimen grips, under both approaches, were registered. The inclined bands of heterogeneous temperature distribution were observed also during the unloading process of the SMA, while the reverse transformation accompanied by temperature decrease took place. Thermomechanical aspects of martensitic and reverse transformations for various strain rates were analyzed under both stress- and strain-controlled tests.     

Solving the Contact Problem for a Rectangular Die on an Elastic Foundation

A method of orthogonal polynomials is proposed to solve the contact problem for a rectangular die on an elastic foundation. For the case of an elastic half-space, an exact formula is derived for the translation of the die under symmetric loading     

Interaction Between an Oscillating Sphere and a Thin Elastic Cylindrical Shell Filled with a Compressible Liquid. Internal Axisymmetric Problem

The problem on the interaction between a spherical body that oscillates in a prescribed manner and a thin elastic cylindrical shell filled with an ideal compressible liquid is formulated. It is assumed that the geometrical center of the sphere is located on the cylinder axis. The problem is solved based on the possibility of representing a partial solution of the Helmholtz equation written in cylindrical coordinates in terms of partial solutions in spherical coordinates, and vice versa. By satisfying the boundary conditions on the surfaces of the sphere and the shell, we obtain an infinite system of linear algebraic equations to determine the coefficients of expansion of the liquid-velocity potential into a Fourier series in terms of Legendre polynomials. The hydrodynamic characteristics of the liquid filling the cylindrical shell are determined and compared with the cases where a sphere oscillates in an infinite liquid and in a rigid cylindrical vessel     

A Boundary Value Problem for an Infinite Elastic Strip with a Semi-infinite Crack

In this paper we study a boundary value problem for an infinite elastic strip with a semi-infinite crack. By using the single and double layer potentials this problem is reduced to a singular integral equation, which is uniquely solved in the Hölder spaces by the Fredholm alternative.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

On a Displacement Based Solution to an Antiplane Crack Problem for Inhomogeneous Anisotropic Elastic Materials

This paper employs a displacement based method to examine an antiplane crack problem for an inhomogeneous elastic material in which the elastic moduli vary continuously with the spatial coordinates. Expressions for the crack tip stress intensity factors and the crack displacement are obtained in terms of Chebyshev polynomials. Numerical results are obtained for some particular inhomogeneous elastic materials.     

文克勒地基上弹性薄板受有限长弹性杆冲击的动力分析

对具有初速的有限长弹性杆与文克勒地基上弹性薄板的冲击问题进行了研究，用伽辽金原理求出了冲击力及薄板位移响应的近似公式，并对计算结果作了分析讨论     

Contact Problem for an Elastic Ring Punch and a Half-Space with Initial (Residual) Stresses*

International Applied Mechanics - The problem of contact interaction without friction between an elastic cylindrical ring punch and an elastic half-space with initial (residual) stresses under...     

Axisymmetric Interaction Problem for a Sphere Pulsating Inside an Elastic Cylindrical Shell Filled with and Immersed Into a Liquid

An interaction problem is formulated for a spherical body oscillating in a prescribed manner inside a thin elastic cylindrical shell filled with a perfect compressible liquid and submerged in a dissimilar infinite perfect compressible liquid. The geometrical center of the sphere is on the cylinder axis. The solution is based on the possibility of representing the partial solutions of the Helmholtz equations written in cylindrical coordinates for both media in terms of the partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions on the sphere and shell surfaces results in an infinite system of linear algebraic equations. This system is used to determine the coefficients of the Fourier-series expansions of the velocity potentials in terms of Legendre polynomials. The hydrodynamic characteristics of both liquids and the shell deflections are determined. The results obtained are compared with those for a sphere oscillating on the axis of an elastic cylindrical shell filled with a compressible liquid (the ambient medium being neglected).     

Dynamics of Phase Transitions and Hysteresis in a Viscoelastic Ericksen's Bar on an Elastic Foundation

This work is a follow-up on the study [32] of interface dynamics and hysteresis in materials undergoing solid-solid phase transitions. We consider the dynamics of a viscoelastic bar with a nonmonotone stress-strain relation and viscous stress linearly proportional to the strain rate. The bar is placed on an elastic foundation with stiffness β mimicking the interaction of phases in higher dimensions. Time-dependent displacement-controlled loading of the bar results in a tilted and serrated hysteresis loop, in qualitative agreement with some experimental observations in shape-memory alloys. The model exhibits three phase transition processes: phase nucleation, interface slip and phase annihilation. Between these dynamic processes the system gets stuck in local minimizers of the potential energy. As β increases from zero, a slip-dominated solution behavior transforms to the one where slip and annihilation events are preceded by a step-by-step nucleation process. We show that this transition is caused by an interplay between the slip-favoring inertia term and the nucleation-favoring elastic foundation terms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Effect of Texture on the Grain-Size-Dependent Functional Properties of NiTi Shape Memory Alloys and Texture Gradient Design:A Phase Field Study

Texture is inevitably introduced during the manufacturing of most NiTi shape memory alloys(SMAs),and the textured nanocrystalline NiTi has been extensively employed in engineering.However,the effect of texture,and the joint effect of grain size(GS)and texture on the functional properties of NiTi SMAs and the corresponding microscopic mechanisms have not been clearly understood yet.In this work,based on the phase field method,the effect of texture on the GS-dependent functional properties of NiTi SMAs,including super-elasticity(SE),one-way shape memory effect(OWSME),and stress-assisted two-way shape memory effect(SATWSME),is investigated,and the corresponding microscopic mechanisms are revealed.Moreover,the samples with discrete geometrical gradients and/or texture gradients are designed to achieve graded functional properties.The simulation results indicate that the dependence of functional properties on texture is due to the effect of crystallographic orientation on martensite transformation and reorientation,which can lead to different inelastic strains.In the designed samples with texture gradients,the stress-strain responses of sheets with various textures are different,allowing for the coordination of overall deformation of the sample by combining such sheets,with varying inelastic deformation degrees.Thus,the overall response of the sample differs from that without texture gradient,leading to the achievement of graded functional properties.The simulation results and new findings in this work contribute to a deeper understanding of the effects of texture,GS,and their interaction on the functional properties of SMAs,and provide valuable reference for the design and development of SMA-based devices with desired functional properties.     

Construction of the Exact Solution to a Torsion Problem for an Elastic Shaft of Variable Cross Section

The exact solution is constructed to a torsion problem for a circular elastic shaft in a medium referred to a spherical coordinate system. One end of the shaft is rigidly fixed and the other is subjected to either tangential forces or a torque. New integral transforms are obtained to solve the problem.     

On Cavitation, Configurational Forces and Implications for Fracture in a Nonlinearly Elastic Material

Experiments on polymers indicate that large tensile stress can induce cavitation, that is, the appearance of voids that were not previously evident in the material. This phenomenon can be viewed as either the growth of pre-existing infinitesimal holes in the material or, alternatively, as the spontaneous creation of new holes in an initially perfect body. In this paper our approach is to adopt both views concurrently within the framework of the variational theory of nonlinear elasticity. We model an elastomer on a macroscale as a void-free material and, on a microscale, as a material containing certain defects that are the only points at which hole formation can occur. Mathematically, this is accomplished by the use of deformations whose point singularities are constrained. One consequence of this viewpoint is that cavitation may then take place at a point that is not energetically optimal. We show that this disparity will generate configurational forces, a type of force identified previously in dislocations in crystals, in phase transitions in solids, in solidification, and in fracture mechanics. As an application of this approach we study the energetically optimal point for a solitary hole to form in a homogeneous and isotropic elastic ball subject to radial boundary displacements. We show, in particular, that the center of the ball is the unique optimal point. Finally, we speculate that the configurational force generated by cavitation at a non-optimal material point may be sufficient to result in the onset of fracture. The analysis utilizes the energy-momentum tensor, the asymptotics of an equilibrium solution with an isolated singularity, and the linear theory of elasticity at the stressed configuration that the body occupies immediately prior to cavitation. This revised version was published online in June 2006 with corrections to the Cover Date.