## 2 Receding Contact of a Beam on Elastic Foundation
### Non-Adhesive Receding Contact  
We consider a uniform straight beam with length \(2l\), width \(D\), height \(h\), and Young modulus \(E\). The flexural rigidity \(EI\) of the beam is given by Equation (1).  
\[EI=EDh^{3}/12 \tag{1}\]  
The beam is resting on an elastic foundation with linear stiffness density \(\alpha\) (stiffness per length of the foundation). A force \(F\) is applied to the beam, as shown in Figure 1. Application of the force causes contact in the region \(-a<x<a\) and separation elsewhere (it is assumed here that the half-length of the beam is larger than the half-width of the contact region).  
Figure 1: A beam resting on an elastic foundation subjected to a concentrated force.  
Let us denote the vertical displacement of the beam at the coordinate \(x\) by \(w(x)\). The deflection line of the beam is determined by the following equation:  
\[\frac{\mathrm{d}^{4}w(x)}{\mathrm{d}x^{4}}+\kappa^{4}w(x)=0 \tag{2}\]  
with  
\[\kappa^{4}=\frac{\alpha}{EI} \tag{3}\]  
In the non-adhesive case considered here, the boundary conditions at the point \(x=a\) read as follows:  
\[w(a)=0 \tag{4}\]  
\[w^{\prime\prime}\left(a\right)=0,\ w^{\prime\prime\prime}\left(a\right)=0 \tag{5}\]  
meaning that at the last contact point, the displacement, the force moment, and the normal force vanish. At the point \(x=0\),  
\[w^{\prime}(0)=0,\ -EIw^{\prime\prime\prime}\left(0\right)=-F/2 \tag{6}\]  
The general solution of Equation (2) reads as follows:  
\[w(x)=A_{1}e^{\lambda_{1}x}+A_{2}e^{\lambda_{2}x}+A_{3}e^{\lambda_{3}x}+A_{4}e ^{\lambda_{4}x} \tag{7}\]  
where \(\lambda_{1,2,3,4}\) are roots of the characteristic equation as follows:  
\[\lambda^{4}=-\kappa^{4} \tag{8}\]  
Solutions of this equation are given by the following explicit expressions:  
\[\lambda_{1}=\frac{1+i}{\sqrt{2}}\kappa,\ \lambda_{2}=-\frac{1+i}{\sqrt{2}} \kappa,\ \lambda_{3}=\frac{1-i}{\sqrt{2}}\kappa,\ \lambda_{4}=-\frac{1-i}{\sqrt{2}}\kappa \tag{9}\]  
where \(i\) is imaginary unit.  
In the following, we also will need derivatives of Solution (7), as follows:  
\[w^{\prime}(x)=A_{1}\lambda_{1}e^{\lambda_{1}x}+A_{2}\lambda_{2}e^{\lambda_{2} x}+A_{3}\lambda_{3}e^{\lambda_{3}x}+A_{4}\lambda_{4}e^{\lambda_{4}x} \tag{10}\]  
\[w^{\prime\prime}\left(x\right)=A_{1}\lambda_{1}^{2}e^{\lambda_{1}x}+A_{2} \lambda_{2}^{2}e^{\lambda_{2}x}+A_{3}\lambda_{3}^{2}e^{\lambda_{3}x}+A_{4} \lambda_{4}^{2}e^{\lambda_{4}x} \tag{11}\]  
\[=i\left(A_{1}e^{\lambda_{1}x}+A_{2}e^{\lambda_{2}x}-A_{3}e^{\lambda_{3}x}-A_{4 }e^{\lambda_{4}x}\right) \tag{12}\]  
Substituting Equations (7) and (10)-(12) into boundary conditions (5) and (6) and solving the resulting system of four linear equations leads to the following equation for the displacement at the contact edge:  
\[w(a)=\frac{F\sqrt{2}\cos\left(\frac{xa}{\sqrt{2}}\right)\cosh\left(\frac{xa}{ \sqrt{2}}\right)}{\kappa^{3}EI\left(\sinh\left(\sqrt{2}xa\right)+\sin\left( \sqrt{2}xa\right)\right)} \tag{13}\]  
Boundary condition (4) can only be fulfilled if \(\cos\left(xa/\sqrt{2}\right)=0\), thus leading to the following solution:  
\[a=\frac{\pi}{\sqrt{2}}\kappa^{-1}=\pi\sqrt[4]{\frac{EI}{4\pi}} \tag{14}\]  
Note that in the non-adhesive case, the size of the contact does not depend on the magnitude of applied force, which is a general feature of receding contact [2].