Fracture Mechanics and Adherence of Viscoelastic Solids

Abstract

Contact of two elastic solids is treated as a thermodynamic problem. It is shown that U = U_E+U_S and U_E = U_S+U_P+U_S are thermo-dynamic potentials respectively for transformations at fixed grips and at fixed load conditions (U_E, U_P, U_S are the elastic, potential, interface energies). Equations giving the displacement δ and the strain energy release rate G as a function of the contact area A and the load P appear to be the equations of state of the system. Two bodies in contact on an area A are in equilibrium if G=w, where w is the thermodynamic (or Dupre’s) work of adhesion. This equilibrium is stable if ∂G/∂A is positive, unstable if negative. The quasistatic force of adherence is the load corresponding to ∂G/∂A = o. But equilibrium may be stable at fixed grips and unstable at fixed load, so that the quasistatic force of adherence may depend on the stiffness of the measuring apparatus. When G>w, the separation of the two bodies starts, and can be seen as the propagation of a crack in mode I. G-w is the force applied to unit length of crack; under this force, the crack takes a limiting speed v, which is a function of the temperature, and one can write

$$G - w = w\phi \left( {{a_T}v} \right).$$

The second term is the drag due to viscoelastic losses at the crack tip and is proportional to was proposed by Gent and Schultz, and Andrews and Kinloch. The function Φ is a characteristic of the material (most probably linked to the frequency dependence of E′ and E″, the real and imaginary part of the Young modulus) and is independent of the geometry and loading system. In this proposed formula surface properties and viscoelastic losses are clearly decoupled from elastic properties and loading conditions that appear in G. If Φ is known, this equation allows one to predict any feature such as kinetics of detachment at fixed load, fixed grips or fixed cross-head velocity δ. (This last point completely solves the problem of tackiness). The only hypotheses are that failure is an adhesive failure and that viscoelastic losses are limited to the crack tip; this last condition means that gross displacements must be elastic for G to be valid in kinetic phenomena.

Three geometries are investigated: adherence of punches, adherence of spheres and peeling. The variation of energies with area of contact is given, and the kinetics of crack propagation under various conditions is studied. Experiments on the adherence of polyurethane to glass confirm the theoretical predictions with a high precision.