刚度调制弹性曲线的分叉
Bifurcation of elastic curves with modulated stiffness
摘要：
我们研究了固定长度的封闭平面弹性曲线的平衡构型，其刚度，也称为弯曲刚度，取决于一个附加的密度变量。基本的变分模型依赖于弯曲能量的形状和密度方面的最小化，可以被认为是一个一维的模拟的Canham-Helfrich模型的异质生物膜。我们提出了一个广义Euler-Bernoulli弹性泛函具有密度依赖的刚度系数。为了处理固有的非凸性的问题，我们引入了一个额外的长度尺度的模型通过密度梯度项。我们推导了Euler-Lagrange方程组，并研究了解关于模型参数的分支结构。分析和数值结果。
英文摘要：
We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness, also known as the bending rigidity, depends on an additional density variable. The underlying variational model relies on the minimization of a bending energy with respect to shape and density and can be considered as a one-dimensional analogue of the Canham-Helfrich model for heterogeneous biological membranes. We present a generalized Euler-Bernoulli elastica functional featuring a density-dependent stiffness coefficient. In order to treat the inherent nonconvexity of the problem we introduce an additional length scale in the model by means of a density gradient term. We derive the system of Euler-Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. Both analytical and numerical results are presented.
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