循环加载下具有塑性的格构弹簧模型响应的镇定
Stabilization of the response of cyclically loaded lattice spring models with plasticity
摘要：
本文建立了一个分析框架，设计应力控制和位移控制的T-周期载荷，使准静态演化的一维弹塑性弹簧网络收敛到一个独特的周期制度。这样一个演化问题的解是一个函数t->（e（t），p（t）），其中e_i（t）和p_i（t）是弹簧i的弹性和塑性变形，由初始条件（e（t0），p（t0））定义在[t0，\infty）上。在我们严格地将问题转化为在d维向量空间E中具有移动多面体C（t）的Moreau扫描过程之后，很自然地期望（基于Krejci的结果）解t->（e（t），p（t））总是收敛于T-周期函数。本文的成果是发现了一类Krejci极限不依赖于初始条件（e（t0），p（t0））的载荷，因此所有的轨迹都接近于相同的T周期区域。所提出的这类扫描过程是运动多面体C（t）的任意d个不同刻面的法向量线性无关的扫描过程。我们进一步将此几何条件与给定弹簧网络的机械特性联系起来。我们发现，当位移控制载荷的数目比给定弹簧网络的节点数少2，且应力控制载荷的大小足够大（但允许）时，运动多面体C（t）的任意d个不同面的法向量是线性无关的。其结果可以看作是弹塑性系统高增益控制方法的一个模拟。在塑性连续介质理论中，相应的结果被称为弗雷德里克-阿姆斯特朗定理。
英文摘要：
This paper develops an analytic framework to design both stress-controlled and displacement-controlled T-periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t-> (e(t),p(t)), where e_i(t) and p_i(t) are the elastic and plastic deformations of spring i, defined on [t0,\infty) by the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the solution t->(e(t),p(t)) always converges to a T-periodic function. The achievement of this paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.
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