penny_GM from An axisymmetric problem for a penny-shaped crack under the influence of the Steigmann–Ogden surface energy

A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann–Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution to this system can be obtained by expanding the unknown functions into the Fourier–Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behaviour of the material system. In particular, the presence of surface energy leads to the size-dependency of the solutions and smoother behaviour of the solutions near the tip of the crack.

本文研究无限大均质各向同性弹性介质中纳米尺度币形裂纹（penny-shaped fracture）的力学问题：该裂纹通过在其表面施加轴对称法向面力实现张开，且裂纹边界上存在Steigmann–Ogden形式的表面能（surface energy）。本文采用由若干未知函数的Hankel变换（Hankel transform）构成的布辛涅斯克势（Boussinesq potential）求解该问题，借助上述未知函数可将原问题转化为包含两个奇异积分微分方程的方程组。通过将未知函数展开为傅里叶-贝塞尔级数（Fourier–Bessel series）可求得该方程组的数值解，随后通过求解线性代数方程组即可得到未知函数的近似解。本文对该数值求解方法的精度进行了分析，针对不同表面能参数取值设置了多组数值算例，并开展了解答对系统力学与几何参数依赖性的参数化研究。结果表明，表面参数对材料系统的力学行为具有显著影响，具体而言，表面能的存在使得解答具有尺寸依赖性，且能使裂纹尖端附近的解答行为更为平滑。