#### Installation — business terrible - 1 part

September 8th, 2015

A slightly exceptional case in LEPM is that shown in Fig. 3.2.4a, where a subsurface (delamination) crack may grow due to buckling of the layer above it, induced by a remotely applied compressive stress oa0- In this case the computation of Q must take into account the geometrical nonlinearity implied in buckling.

To do this, one computes the elastic energy of a buckled beam, 14, as the work supply when no dissipative processes take place. This coincides with the area swept by the P-и curve, where P is the compressive load applied to the beam and и is the beam shortening. When h, b, and 2a are, respectively, the depth, thickness, and length of the beam, the P-n curve coincides with the straight line P = bhEu/2a for loads below Euler’s buckling load Рц and is horizontal (P — Pe) for further displacements (Fig. 3.2.4). The

area under the curve Ub is

Ыь = Peu – – Pe ue = labhtTE (— – (3.2.13)

where oe — Pp/bh is the buckling stress, which, for fixed ends and rectangular cross-section, is

The fundamental simplification in our problem consists of assuming that the displacement of the ends of the buckling layer of length 2a is imposed by the deformation of the surrounding material which stays at stress level (7qq. So we have

u = 2a~ (3.2.15)

jь

and, after substitution into Eq. (3.2.13), the strain energy of the buckled layer is

ctf

Ub = abh—(2/700 – aE) (3.2.16)

The energy U of the whole system is Ub plus the strain energy of the surrounding material, which is the strain energy density rjj2E times the surrounding volume, equal, in turn, to the total (constant) volume of the body V minus the volume of the buckling layer 2abh. The resulting expression is

U = (V – 2aWi)|| + obh—~(2cjca ~ *s) = v|| – (3.2.17)

From this, the expression for Q follows at once using Eq. (2.1.15) with the condition, following from the simplifications used in the derivation, that cr^ remains constant at constant displacement. After inserting Eq. (3.2.14) and differentiating with respect to a, one gets the following expression for the energy release rate:

This result captures some, but not all, of the important aspects of the problem of delamination in layered composites (Sallam and Simitses 1985, 1987; Yin, Sallam, and Simitses 1986).