Solid Mechanics

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Point contact on a 2D half-plane

A starting point to solve problems in contact mechanics is to understand the effect of a point-load applied to a homogeneous, linear elastic, isotropic half-plane. This problem may be defined either as plane stress or plain strain (for solution with FreeFem++ we have choosen the latter). The traction boundary conditions for this problem are: \begin{equation} \sigma_{xy}(x,0) = 0, \quad \sigma_{yy}(x,y) = -P\delta(x,y) \end{equation} where $\delta(x,y)$ is the Dirac delta function. Together these boundary conditions state that there is a singular normal force $P$ applied at $(x,y) = (0,0)$ and there are no shear stresses on the surface of the elastic half-plane.

The analytical relations for the stresses can be found from the Flamant solution (stress distributions in a linear elastic wedge loaded by point forces a the tip. When the "wedge" is flat we get a half-plane. The derivation uses polar coordinates.) and are given as: \begin{equation} \sigma_{xx} = -\frac{2P}{\pi} \frac{x^2y}{\left(x^2+y^2\right)^2}, \end{equation} \begin{equation} \sigma_{xy} = -\frac{2P}{\pi} \frac{y^3}{\left(x^2+y^2\right)^2}, \end{equation} \begin{equation} \sigma_{yy} = -\frac{2P}{\pi} \frac{xy^2}{\left(x^2+y^2\right)^2}, \end{equation} for some point $(x,y)$ in the half-plane. From this stress field the strain components and thus the displacements can be determined. The displacements are given by \begin{equation} u = -\frac{P}{4\pi\mu}\left((\kappa-1)\theta - \frac{2xy}{r^2}\right), \end{equation} \begin{equation} v = -\frac{P}{4\pi\mu}\left((\kappa+1)\log r - \frac{2y^2}{r^2}\right), \end{equation} where $$r = \sqrt{x^2+y^2}$$ and $$\tan \theta = \frac{x}{y}.$$ The symbol $\kappa$ is known as Dundars constant as is defined as \[ \kappa = \begin{cases} 3 - 4\nu & \qquad \text{plane strain}, \\ \cfrac{3 - \nu}{1+\nu} & \qquad \text{plane stress}. \end{cases} \]