Stress equilibrium relations and boundary conditions

The stress equilibrium relations for such a film in plane strain approximation is written in terms of the pressure field as,

,

(19.1)

Where, and are the deformation components along and directions respectively and is the shear modulus of the adhesive material. We can use the plane strain approximation because the film is infinitely long along the direction and the deformations remain independent of . The double derivatives of displacements represented by define the strain gradients. The incompressibility condition is written as

(19.2)

These equations are solved using the following boundary conditions:

(a) Displacements along and are zero at the film substrate interface:

(19.3a)

(b) The normal stress is continuous at the interface of the film and the plate, so that

for < 0

(19.3b)

for < 0

(19.3c)

For small bending of plate in one dimension, the normal stress is equal to the bending stress which can be expressed in terms of the stiffness or flexural rigidity D of plate as

Here is the vertical displacement of the film at and is the distance of the line of application of the peeling force from the contact line at .

We can simplify the above set of equations and boundary conditions because of large separation of characteristic length-scales along the and directions respectively , i.e. and : .
For example, . Therefore, we can neglect all terms except , so that the equations of equilibrium simplify to

and

(19.4)