Stabilization of the response of cyclically loaded lattice spring models with plasticity
Stabilization of the response of cyclically loaded lattice spring models with plasticity
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DOI：
10.1051/cocv/2020043
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发表时间：
2017-08
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通讯作者：
I. Gudoshnikov;O. Makarenkov
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文献类型：
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作者：
I. Gudoshnikov;O. Makarenkov
本文构建了一个分析框架，用于设计应力控制和位移控制的T周期载荷，使得弹塑性弹簧的一维网络的准静态演化收敛到一个唯一的周期状态。此类演化问题的解是一个函数\(t\mapsto(e(t),p(t))\)，其中\(e_i(t)\)是弹簧\(i\)的弹性伸长量，\(p_i(t)\)是弹簧\(i\)的松弛长度，由初始条件\((e(t_0),p(t_0))\)在\([t_0,\infty)\)上定义。在我们将问题严格转化为在维度为\(d\)的向量空间\(E\)中具有移动多面体\(C(t)\)的莫罗扫掠过程之后，基于克雷伊奇（Krejci）的一个结果，很自然地期望（当\(t\rightarrow\infty\)时）弹性分量\(t\mapsto e(t)\)总是收敛到一个\(T\)周期函数。本文的成果在于发现了一类载荷，对于这类载荷，克雷伊奇极限不依赖于初始条件\((e(t_0),p(t_0))\)，因此所有轨迹都趋近于相同的\(T\)周期状态。所提出的扫掠过程类别是移动多面体\(C(t)\)的任意\(d\)个不同面的法向量线性无关的类别。我们进一步将此几何条件与给定弹簧网络的力学性能联系起来。我们发现，如果位移控制载荷的数量比给定弹簧网络的节点数量少两个，并且当应力控制载荷的大小足够大（但可允许）时，移动多面体\(C(t)\)的任意\(d\)个不同面的法向量是线性无关的。该结果可被视为弹塑性系统的高增益控制方法的类似物。在连续塑性理论中，相应的结果被称为弗雷德里克 - 阿姆斯特朗定理。
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 ei(t) is the elastic elongation and pi(t) is the relaxed length of spring i, defined on [t0, ∞) 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 elastic component t↦e(t) always converges to a T-periodic function as t →∞. 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 normals 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 normals 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.