A continuum thermodynamics formulation for micro-magneto-mechanics with applications to ferromagnetic shape memory alloys

A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the evolution of magnetic domain and martensite twin structures in ferromagnetic shape memory alloys. The theory falls into the class of phase-field or diffuse-interface modeling approaches. In addition to the standard mechanical and magnetic balance laws, two sets of micro-forces and their associated balance laws are postulated; one set for the magnetization order parameter and one set for the martensite order parameter. Next, the second law of thermodynamics is analyzed to identify the appropriate material constitutive relationships. The proposed formulation does not constrain the magnitude of the magnetization to be constant, allowing for spontaneous magnetization changes associated with strain and temperature. The equations governing the evolution of the magnetization are shown to reduce to the commonly accepted Landau-Lifshitz-Gilbert equations for the case where the magnetization magnitude is constant. Furthermore, the analysis demonstrates that under certain limiting conditions, the equations governing the evolution of the martensite-free strain are shown to be equivalent to a hyperelastic strain gradient theory. Finally, numerical solutions are presented to investigate the fundamental interactions between the magnetic domain wall and the martensite twin boundary in ferromagnetic shape memory alloys. These calculations determine under what conditions the magnetic domain wall and the martensite twin boundary can be dissociated, resulting in a limit to the actuating strength of the material.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

Large deformation of anisotropic austenitic stainless steel sheets at room temperature: Multi-axial experiments and phenomenological modeling

A newly developed multi-axial testing technique for sheet materials is employed to investigate the inelastic response of a temper-rolled stainless steel 301LN under isothermal quasi-static loading conditions at room temperature. The experimental technique consists of a flat sheet specimen, which is subject to combinations of shear and normal loading using a custom-made dual-actuator system. The large deformation behavior under monotonic loading is determined along more than 20 distinct radial paths in the stress space. The experimental results indicate that Hill's quadratic yield function along with an associated flow rule provides a good approximation of the initial yield behavior of this anisotropic two-phase FCC/BCC sheet material. Based on the experimental data for radial monotonic loading, it is concluded that conventional isotropic-kinematic hardening models cannot successfully describe the strain hardening of this austenitic steel. Instead, a non-associated anisotropic hardening model is proposed that relates the increase in yield strength to an isothermal martensitic transformation kinetics law. The comparison of the model predictions with the experimental results shows very good agreement for all biaxial and uniaxial experiments.     

A numerical analysis of flat internal cracks under mixed mode loading

A numerical/analytical approach is proposed to determine the stress intensity factors K_I, K_II, and K_III of a 3D internal crack. The main point of this approach is the meshing technique that can model very sharp crack fronts. The meshing technique is based on an elliptical coordinate transformation that starts from a circular crack. It allows the obtainment of a curved crack front with elements normal to the crack front. Remarkable accuracy can be obtained for elliptical crack fronts with axes ratio smaller that 0.01. Accuracy demonstration is provided for cylindrical element with an inclined internal crack subjected to uni-axial tension. This case corresponds to crack propagation for all three modes of loading, the solution of which can checked with references’ results.     

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

Falkner-Skan方程的近似解析解

研究了粘性流体绕流楔型物体的Falkner-Skan边界层方程求解问题.利用Adomian拆分方法,通过引入Crocco变量变换将无穷区间的边界值问题转为初值问题并利用Padé逼近技巧确定初值,给出了一种有效的解析分解方法.进一步,本文设计了一种数值解法,将本文得到的近似解析解及数值结果与早期研究者Hartree等人的结果进行了比较,证明了本文提出的解法的有效性和可靠性.     

Application of nonsmooth transformations to analyze a vibroimpact duffing system

The forced vibrations of an impact Duffing oscillator with one degree of freedom are studied using Zhuravlev's nonsmooth transformation and the multiple-scales method. Detailed consideration is given to the forced vibrations in the domain of resonance family. The stability and bifurcations of periodic vibrations are analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 114–121, October 2008.     

Lax Connection of Strings in γ-Deformed Backgrounds

In this paper we consider classical strings propagating in γ-deformed AdS3 × S^3 backgrounds generated by TsT transformation on the AdS3 sector, which is described as the group manifold SL（2, R）, then we prove that the U（1） currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds. Using TsT transformation, we can derive the local Lax connection and the monodromy matrix in γ-deformed backgrounds with the spectral parameter, which ensures the classical integrability of the string theories.     

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

In this paper, we put our focus on a variable-coe~cient fifth-order Korteweg-de Vries （fKdV） equation, which possesses a great number of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived, such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.     

KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.