论文标题
机械和自我引力力下球体的空化
Cavitation of a spherical body under mechanical and self gravitational forces
论文作者
论文摘要
在本文中，考虑到人体本身引起的引力场的作用，我们将寻找各向同性可压缩弹性的能量功能的最小化。我们考虑到身体的外边界遇到Dirichlet类型边界条件的位移问题。对于占用单位球$ \ MATHCAL {B} \ in \ MATHBB {R}^3 $的球形对称体，最小化是在径向对称变形的类别中完成的。我们给出了存在此类最小化器的条件，以满足欧拉 - 拉格朗日的满足，并表明，对于大型位移，最小化器必须在中心发展一个腔。给出了一种用于近似这些最小化器的数值方案，并与一些模拟一起显示了腔半径的依赖性和最小能量对人体的位移和质量密度的依赖性。
In this paper we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider the displacement problem in which the outer boundary of the body is subjected to a Dirichlet type boundary condition. For a spherically symmetric body occupying the unit ball $\mathcal{B}\in\mathbb{R}^3$, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler--Lagrange equations, and show that for large displacements the minimizer must develop a cavity at the centre. A numerical scheme for approximating these minimizers is given together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.
