## 2 Receding Contact of a Beam on Elastic Foundation
### Adhesive Receding Contact  
Consider now an infinite beam resting on the foundation and adhering to it. We consider two cases: At first, we assume that the initial state of the beam is a partial contact resembling the non-adhesive case (Figure 2). In this case, adhesion will only change the contact size. Later, we will assume that the initial state of the beam is the full contact. In this case, we will identify the position at which the beam starts to detach from the foundation. In the first case, all equations of the previous section remain valid with the exception of boundary condition (4), which now reads  
\[w(a)=-\Delta l_{c} \tag{15}\]  
where \(\Delta l_{c}\) is the critical elongation at which individual springs at the edge of contact will detach. This critical length can be obtained from the Griffith energy balance [22]. In the moment of detachment of an edge spring, the elastic energy stored in the spring is equal to \(\Delta U_{\text{el}}=\frac{a\Delta x}{2}\Delta l_{c}^{2}\). According to the principle of virtual work, the spring is in equilibrium if this energy is equal to the work of adhesion \(\Delta U_{\text{adh}}=\gamma\Delta A=\gamma D\Delta x\), where \(\gamma\) is the specific work of adhesion, \(D\) is the width of the beam/foundation, and \(\Delta x\) is the spacing between adjacent springs. This leads to the following result:  
\[\Delta l_{c}=\sqrt{\frac{2\gamma D}{\alpha}} \tag{16}\]  
For a beam on an elastic foundation, this length does not depend on the size of the system or on the position of the spring. Note that this applies also to a beam resting on a thin elastic layer [23]. In a non-adhesive contact, the contact size \(a\) is determined by the intersection of the curve in Figure 3 with the \(x\)-axis. The only difference from the adhesive case is that now, the size of the contact is determined by the intersection of the \(w(a)\) curve with the horizontal line at the negative height \(-\Delta l_{c}\). Obviously, the presence of adhesion leads to an increase in the contact size. The \(w(a)\) curve has a minimum \(w_{\text{min}}=-0.0974\frac{F}{x^{3}EI}\) at \(a=3.3829\kappa^{-1}\). If the critical elongation \(\Delta l_{c}\) is larger than \(|w_{\text{min}}|\), Equation (15) has no solutions, and the beam jumps into contact along its entire length. This means that if we start with an open contact and decrease the normal force \(F\), the contact area increases continuously up to the point of instability, where the beam suddenly comes into complete contact. Now, let us consider the beam which initially is in contact at its full length. The boundary conditions in this case are given by \(w(\infty)=0\), \(w^{\prime}(\infty)=0\), \(\ldots\) This means that the terms with \(A_{1}\) and \(A_{3}\) must vanish, as corresponding exponents have positive real parts. Thus, the solution reads as follows:  
\[w(x)=A_{2}e^{\lambda_{2}x}+A_{4}e^{\lambda_{4}x} \tag{17}\]  
Figure 2: A beam resting on an elastic foundation and subjected to a concentrated force in the presence of adhesion. The beam is shown in the state of an incomplete contact. The springs at the contact edge have a critical elongation \(\Delta l_{c}\). and boundary conditions (6) take the following form:  
\[w^{\prime}(0)=A_{2}\lambda_{2}+A_{4}\lambda_{4}=0 \tag{18}\]  
\[w^{\prime\prime\prime}\left(0\right)=i(A_{2}\lambda_{2}-A_{4}\lambda_{4})=\frac {F}{2 EI} \tag{19}\]  
Solving this system of linear equations and substituting the result into (17) gives the following:  
\[w(x)=\frac{F}{2\sqrt{2}EI\kappa}e^{-\frac{\kappa x}{\sqrt{2}}}\biggl{(}\sin \biggl{(}\frac{\kappa x}{\sqrt{2}}\biggr{)}+\cos\biggl{(}\frac{\kappa x}{ \sqrt{2}}\biggr{)}\biggr{)} \tag{20}\]  
The dependence (20) is shown in Figure 4. One can see that the displacement is positive (which corresponds to positive pressure) up to \(x=3.33\kappa^{-1}\) and becomes negative for larger coordinates, which corresponds to tensile stresses. The contact will be lost at the point with the highest tensile stresses, which is at the point \(x=4.44\kappa^{-1}\).