Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.     

A combined analytical,numerical, and experimental study of shape-memory-alloy helical springs

In this paper, the pseudoelastic response of shape memory alloy (SMA) helical springs under axial force is studied both analytically and numerically. In the analytical solution two different approximations are considered. In the first approximation, both the curvature and pitch effects are assumed to be negligible. This is the case for helical springs with large ratios of mean coil radius to the cross sectional radius (spring index) and small pitch angles. Using this assumption, analysis of the helical spring is reduced to that of the pure torsion of a straight bar with circular cross section. A three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. In the next step, the curvature effect is included and the SMA helical spring is analyzed using the exact solution presented for torsion of curved SMA bars. In this refined solution, the effect of the direct shear force is also considered. In the numerical analyses, the three-dimensional constitutive equations are implemented in a finite element method and using solid elements the loading–unloading of an SMA helical spring is simulated. Analytical and numerical results are compared and it is shown that the solution based on the SMA curved bar torsion gives an accurate stress analysis in the cross section of the helical SMA spring in addition to the global load–deflection response. All the results are compared with experimental data for a Nitinol helical spring. Several case studies are presented using the proposed analytical and numerical solutions and the effect of changing different parameters such as the material properties and temperature on the loading–unloading hysteretic response of SMA helical springs is studied. Finally, some practical recommendations are given for improving the performance of SMA helical springs used as energy dissipating devices, for example for seismic applications.     

On the torsion problem of strain gradient elastic bars

The torsion problem of elastic bars of any cross-sections is discussed, into the context of strain gradient elasticity. It is proven that torsion problem is feasible only for the bars with circular cross-sections. For the other bars (with non-circular cross sections), the non-classical boundary conditions are not satisfied.     

Dynamic multi-axial behavior of shape memory alloy nanowires with coupled thermo-mechanical phase-field models

The objective of this paper is to provide new insight into the dynamic thermo-mechanical properties of shape memory alloy (SMA) nanowires subjected to multi-axial loadings. The phase-field model with Ginzburg–Landau energy, having appropriate strain based order parameter and strain gradient energy contributions, is used to study the martensitic transformations in the representative 2D square-to-rectangular phase transformations for FePd SMA nanowires. The microstructure and mechanical behavior of martensitic transformations in SMA nanostructures have been studied extensively in the literature for uniaxial loading, usually under isothermal assumptions. The developed model describes the martensitic transformations in SMAs based on the equations for momentum and energy with bi-directional coupling via strain, strain rate and temperature. These governing equations of the thermo-mechanical model are numerically solved simultaneously for different external loadings starting with the evolved twinned and austenitic phases. We observed a strong influence of multi-axial loading on dynamic thermo-mechanical properties of SMA nanowires. Notably, the multi-axial loadings are quite distinct as compared to the uniaxial loading case, and the particular axial stress level is reached at a lower strain. The SMA behaviors predicted by the model are in qualitative agreements with experimental and numerical results published in the literature. The new results reported here on the nanowire response to multi-axial loadings provide new physical insight into underlying phenomena and are important, for example, in developing better SMA-based MEMS and NEMS devices     

形状记忆合金作动器的设计及优化

采用Ｂｒｉｓｉｏｎ［１］等提出的ＳＭＡ材料的本构关系，分析了形状记忆合金作动器在三种热交换方式（自然冷却、强制冷却和采用半导体热泵技术）下的动力特性，同时研究了ＳＭＡ丝在恒载和载荷变化时对完成正逆相变所用时间的影响，并简单讨论了影响作动器响应的几个因素和改进方法     

Nonlinear finite element analysis of shape memory alloy (SMA) wire reinforced hybrid laminate composite shells

This study introduces a non-linear finite element analysis approach to the procedure of modeling hybrid laminate composite shells with embedded shape memory alloy (SMA) wire subjected to coupled structural and thermal loading. Numerical analyses of SMA wire reinforced composite laminates were carried out by synergizing the non-linear laminate shell element with Brison's model of the SMA constitutive law. To verify the proposed procedure, the present illustrative applications involve rectangular laminated panels clamped along one side. Analysis results were compared with corresponding experimental results from a prior study. Several test cases that depend on the volume fraction of SMA, temperature, and ply angles are presented to illustrate the highly entangled thermo-mechanical behavior of shape memory alloy hybrid composites (SMAHCs). The results of the numerical analysis show the ability of the suggested procedure to compute the thermo-mechanical behavior of a SMAHC in accordance with the SMA's internal phase transformations induced by stress and temperature variation and demonstrate very good agreement with experimental results.     

Internal loops in superelastic shape memory alloy wires under torsion – Experiments and simulations/predictions

Understanding torsional responses of shape memory alloy (SMA) specimens under partial or fully transformed cases with internal loops is of particular importance as the entire response might not be always utilized and only a portion of the entire response (internal loop) might be of significance to designers. In this work, we present experimental results of large complex loading and unloading torsional cycles which were conducted on superelastic SMA wires, under isothermal conditions with the purpose of elucidating the torsional internal loop response during loading and unloading. Such data hereto has not been available in open literature. Utilizing this data, we model the torsional response of superelastic SMA wires subjected to various loading and unloading situations that can result in different extents of transformation.A thermodynamically consistent Preisach model (Rao and Srinivasa, 2013) captures such complex internal loops with a high degree of precision by modeling driving force for phase transformation vs. volume fraction of martensite relationships. This approach is different from capturing purely phenomenological stress–strain or stress–temperature Preisach models. The thermodynamic approach utilized here has broader predictive capability. The model predictions indicate good agreement with the internal loop structures even though only the outer loop information was used for model calibration. The addition of a single inner loop information for model calibration greatly improves the predictions.     

The warping functions of regular prismatic bars under torsion evaluated by caustics

Reflection of a bundle of coherent light on the warped cross section of a prismatic bar submitted to torsion forms a caustic on a receiver plane. From the mathematical expression of this curve and the theory of reflected caustics, it is possible to evaluate accurately the warping function of the cross section. Using this idea, it was possible to study the torsion problem in prismatic bars with sections which were equilateral triangles and squares. It was observed that the shape of the caustic is an hypocycloid curve with three or four cusps respectively. By evaluating the warping function by using elements from the respective caustics it was possible to find out that, for the triangular cross section, the expression for the warping function coincided exactly with the expression given by the exact solution of the problem. For the square cross section, a closed-form solution for its warping function was readily derived, to which the series approximation solution differed only by a few percent at maximum for the shear stresses. Since the method can be readily extended to any canonical polygonic cross section, it constitutes a general solution for the torsion of prismatic bars, which approximates their exact deformations better than the solutions based on the Saint-Vénant assumptions.     

Analytical series solution for the fully developed forced convection duct flow with frictional heating and variable viscosity

Laminar forced convection flow of a liquid in the fully developed region of a circular duct with isothermal wall is analyzed. The effects of viscous dissipation as well as of temperature dependent viscosity are taken into account. The coupled momentum and energy equations are solved analytically by means of a power series method. Then, reference is made to the Poiseuille model for the temperature change of viscosity. For a fixed value of the axial pressure gradient along the duct, dual solutions are found for the velocity and temperature fields. Although dual solutions correspond to the same value of the axial pressure gradient, they lead in general to different values of the average fluid velocity, of the average fluid temperature and of the wall heat flux. It is shown that, for a given fluid and for a fixed duct radius, the absolute value of the axial pressure gradient has an upper bound above which no steady laminar solution can exist.     

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基于形状记忆合金Brinson一维热力学本构关系和von K\'{a}rm\'{a}n几何非线性薄板理论,研究了径向嵌入SMA丝复合材料加热圆板在横向均布 机械载荷作用下的弯曲响应, 获得了周边不可移简支和夹紧圆板的中心最大挠度与升温之间的关系曲线. 结果表明,形状 记忆合金丝在从马氏体向奥氏体的逆相变过程中所产生的相变回复力对板的弯曲变形具有明 显的调整作用. 通过嵌入SMA纤维丝和施加升温载荷可以主动而有效地调节受机 械载荷作用圆板的弯曲变形.     

Forced Convection with Slip-flow in a Channel or Duct Occupied by a Hyper-porous Medium Saturated by a Rarefied Gas

An analytic solution is obtained for forced convection flow in a parallel-plates channel or a circular duct occupied by a hyper-porous medium saturated with a rarefied gas in the slip-flow regime, for the case of uniform flux boundary conditions. As expected, it is found that velocity slip leads in general to increased heat transfer and temperature slip leads to reduced heat transfer.     

扇形截面杆扭转的应力分析

柱体扭转的基本方程为非齐次偏微分方程,在极坐标系下,利用分离变量法及傅立叶级数展开法,求出了扭转应力函数,进一步即可计算出扇形截面杆在外力偶作用下,扭转角和横截面上剪应力的精确解答。这种方法为精确解法,在各种机械及其他工程设备中,对受扭转作用的扇形截面杆设计,有一定实用价值。     

Comparison of isotropic hardening and kinematic hardening in thermoplasticity

A coupled thermo-mechanical problem is presented in this paper. The constitutive model is based on thermoplastic model for large strains where both kinematic and isotropic hardening are included. It is shown that a non-associated plasticity formulation enables thermodynamic consistent heat generation to be modeled, which can be fitted accurately to experimental data. In the numerical examples the effect of heat generation is investigated and both thermal softening and temperature-dependent thermal material parameters are considered. The constitutive model is formulated such that pure isotropic and pure kinematic hardening yield identical uniaxial mechanical response and mechanical dissipation. Thus, differences in response due to hardening during non-proportional loading can be studied. Thermally triggered necking is studied, as well as cyclic loading of Cook’s membrane. The numerical examples are solved using the finite element method, and the coupled problem that arises is solved using a staggered method where an isothermal split is adopted.     

Lattice Boltzmann simulation of a fluid flow around a triangular unit of three isothermal cylinders

The lattice Boltzmann method is employed to simulate heat transfer in the flow past three arrangements of elliptical and circular cylinders under an isothermal boundary condition. The lattice Boltzmann equations and the Bhatnagar–Gross–Krook model are used to simulate two-dimensional forced convection at 30 ≤ Re ≤ 100 and Pr = 0.71. Pressure distributions, isotherms, and streamlines are obtained. Vortex shedding maps are observed in detail for several cases. The present results are in good agreement with available experimental and numerical data.     

高应力岩体爆破卸荷过程中应变率及应变能特征

针对高应力岩体爆破开挖卸载问题,自制了一台轴向加、卸载实验测试平台,通过实验测试获得了爆破卸荷过程中岩杆的动态应变及应变率数据。实测数据表明:开挖面附近岩体的爆破加、卸载以及初始应力卸载应变率均在10?1 s?1量级以上,验证了高地应力区岩体爆破开挖卸荷是一动态过程。建立了初始应力卸载一维力学模型,揭示了卸载波的传播机制;通过分析爆破卸荷过程应变能密度的时空分布特征,建立了应变能密度与各阶段应变率变化规律的联系。结合实测数据,采用隐式-显式顺序求解方法,进一步分析了高应力区岩体爆破卸荷荷载各阶段应变率沿岩杆的变化规律。结果表明:爆破加载阶段的平均应变率沿杆件逐渐衰减,且衰减速度逐渐减小;爆破卸阶段平均应变率沿杆件也呈衰减趋势;而初始应力的应变能稳定释放,其平均应变率无衰减趋势。     

Effects of matrix inelasticity on the overall hysteretic behavior of TiNi-SMA fiber composites

The effects of the inelastic deformation of the matrix on the overall hysteretic behavior of a unidirectional titanium–nickel shape-memory alloy (TiNi-SMA) fiber composite and on the local pseudoelastic response of the embedded SMA fibers are studied under the isothermal loading and unloading condition. The multiaxial phase transformation of the SMA fibers is predicted using the phenomenological constitutive equations which can describe the two-step deformation due to the rhombohedral and martensitic transformations, and the inelastic behavior of the matrix material using the standard nonlinear viscoplastic model. The average behavior of the SMA composite is evaluated with the micromechanical method of cells. It is observed that the inelastic deformation of the matrix due to prior tension results in a compressive stress in the matrix after unloading of the SMA composite and this residual stress impedes the complete recovery of the pseudoelastic strain of the SMA fibers. This explains that a closed hysteresis behavior of the SMA composite is no longer observed in contrast with the case that an elastic behavior of matrix is assumed. The predicted local stress–strain behavior indicates that the cyclic response of matrix is crucial to the design of the hysteretic performance of the SMA composite under the repeated loading conditions.     

Two scale damage model and related numerical issues for thermo-mechanical High Cycle Fatigue

On the idea that fatigue damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), a three-dimensional two scale damage model has been proposed for High Cycle Fatigue applications. It is extended here to anisothermal cases and then to thermo-mechanical fatigue. The modeling consists in the micromechanics analysis of a weak micro-inclusion subjected to plasticity and damage embedded in an elastic meso-element (the RVE of continuum mechanics). The consideration of plasticity coupled with damage equations at microscale, altogether with Eshelby–Kröner localization law, allows to compute the value of microscopic damage up to failure for any kind of loading, 1D or 3D, cyclic or random, isothermal or anisothermal, mechanical, thermal or thermo-mechanical. A robust numerical scheme is proposed in order to make the computations fast. A post-processor for damage and fatigue (DAMAGE_2005) has been developed. It applies to complex thermo-mechanical loadings. Examples of the representation by the two scale damage model of physical phenomena related to High Cycle Fatigue are given such as the mean stress effect, the non-linear accumulation of damage. Examples of thermal and thermo-mechanical fatigue as well as complex applications on real size testing structure subjected to thermo-mechanical fatigue are detailed.     

An incremental variational approach to coupled thermo-mechanical problems in anelastic solids. Application to shape-memory alloys

We study the coupled thermo-mechanical problem that is obtained by combining generalized standard materials with Fourier’s law for heat conduction. The analysis is conducted in the framework of non-smooth mechanics in order to account for possible constraints on the state variables. This allows models of damage and phase-transformation to be included in the analysis. In view of performing numerical simulations, an incremental thermo-mechanical problem and corresponding variational principles are introduced. Conditions for existence of solutions to the incremental problem are discussed and compared with the isothermal case. The numerical implementation of the proposed approach is studied in detail. In particular, it is shown that the incremental thermo-mechanical problem can be recast as a concave maximization problem and ultimately amounts to solve a sequence of linear thermal problems and purely mechanical (i.e. at a prescribed temperature field) problems. Therefore, using the proposed approach, thermo-mechanical coupling can be implemented with low additional complexity compared to the isothermal case, while still relying on a sound mathematical framework. As an application, thermo-mechanical coupling in shape memory alloys is studied. The influence of the loading strain-rate on the phase transformation and on the overall stress–strain response is investigated, as well as the influence of the thermal boundary conditions. The numerical results obtained by the proposed approach are compared with numerical and experimental results from the literature.     

Control of axisymmetric resonant vibrations and self-heating of shells of revolution with piezoelectric sensors and actuators

The coupled problem of forced vibrations and self-heating of thermoviscoelectroelastic shells of revolution with piezoceramic sensor and actuator under monoharmonic loading is solved. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoelectroelasticity is solved by time-marching integration, using discrete orhogonalization to integrate the equations of elasticity and explicit finite differencing to solve the heat conduction equation. The effect of the dimensions of the sensor and actuator and self-heating on the sensor voltage and on the active damping of forced vibrations of a circular plate under uniform monoharmonic transverse pressure is studied