形状记忆合金梁在随机激励下的随机分岔与首次穿越

根据形状记忆合金（SMA）的等应变拉压实验的数据，利用van der Pol环模型模拟了形状记忆合金在加载和卸载过程中的应力应变迟滞环特性。并且根据弹性理论和Galerkin方法建立了形状记忆合金简支梁在受轴向高斯白噪声激励时的振动模型。应用拟不可积Hamilton随机平均法将函数表示为一维扩散过程后，通过最大Lyapunov指数判断系统的局部稳定性，同时用奇异边界理论讨论了系统的全局稳定性。随后通过分析稳态概率密度和联合概率密度的图形特征，得到了此模型的随机Hopf分岔现象，并讨论了系统产生随机Hopf分岔的条件。最后，用数值法模拟了系统在特定初始条件和边界条件下可靠性函数和首次穿越时间的概率密度函数所满足的BK(Backward Kolmogorov)方程，并且分析了系统发生首次穿越的条件。

Stochastic bifurcation and first-passage failure of shape memory alloy beam subjected to stochastic excitation

Based on the isostrain loading and unloading experimental data of SMA(shape memory alloy),the van der Pol cycle model was used to characterize the hysteretic nonlinearity of material.According to the theory of elastic and Galerkin’s method,the vibration model was built for the beam excited by the Gaussian noise.The Hamilton function was taken to describe the one dimensional diffusion process by using stochastic average method for quasi-non-integrable-Hamiltonian systems.Using the largest Lyapunov exponent and the singular boundary theory simultaneously,both the local and global stability of the system were studied.The stochastic Hopf bifurcation was analyzed by observing the changes of stationary and joint probability densities.Finally,the Backward Kolmogorov equation for the reliability function and the first-passage time was established and the numerical results were given under designated boundary and initial conditions.