Stability of Equilibrium and Adherence Force

The equilibrium given by G = w can be stable, unstable, or neutral. A thermodynamic system under a given constraint is stable if the corresponding thermodynamic potential is minimum, i.e., if its second derivative is positive. Thus, from Eqs. (3) and (4), stability is defined by 

In this last case the stability depends on the stiffness of the testing machine, since elastic energy stored in the spring can be used for crack propagation. 

The quantity (dP/db) = k is the stiffnness of the two elastic solids, and is positive. So (dG/ can be zero while (dG/dA) is still positive. Equation (13) shows that the stability range increases monotonically with the stiffness, from the fixed load case (k = 0) to the fixed grips case (k = oo). Equations (13) and (14) have been obtained by a number of investigators.( - ) If, under a stable equilibrium, a fluctuation decreases A, then G decreases incrementally and one has G w the crack recedes to its equilibrium position. It can only advance if the load P or displacement 8 is varied, bringing back G to the value w one is dealing with controlled rupture of an adhesive joint. In this case one has 

Starting from a stable equilibrium with the two bodies compressed (P 0, 8 0) let us decrease the cross-head displacement A one generally encounters a progressive reduction in the area of contact, i.e., a controlled rupture with (dG/dA) 0 up to a point where (dGfdA) = 0 the equilibrium becomes unstable and the crack extends spontaneously toward the rupture under this given cross-head displacement. It extends with acceleration, for the crack extension force G-w increases as A decreases. 

Equation (15) shows that the limit of stability corresponds to dA = 0, 

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