# On the propensity of laminates to delaminate

## Authors

## Abstract

Laminates have a propensity to delaminate; the mathematical plane between adjacent plies offers a preferred path for crack propagation, irrespective of the nature of the stress field that gives rise to the elastic strain energy released. This is because the plane between plies is characterised by a specific fracture surface energy significantly lower than those for internal surfaces that intersect fibres. In the second of his two classical publications on fracture, A.A. Griffith showed how crack rotation in two-dimensional stress fields occurs. This suggests how, in a laminate, pre-existing flaws are able to seek out the plane of lamination; here, crack rotation under the influence of, for example, shear stress is examined in the context of laminates designed for use in aerospace. One physical consequence of Griffith’s calculation is the prediction of crack propagation in elastic solids subjected to bidimensional compression with strongly unequal principal stresses. A simple bidimensional compression rig has been devised to investigate this prediction. To obviate the risk of delamination, it will be necessary to move away from anisotropic lay-ups, and further develop three-dimensional weaves and methods for weaving three-dimensional weaves. A method whereby a three-dimensional fibre weave, which has cubic symmetry and no zero-valued shear moduli, might be weaved is outlined.

## Introduction

### C.E. Inglis

About 100 years ago, Inglis examined the problem of fracture of ship hulls which, at that time, were made of rivoted steel plate. Along the crack path, he noticed that, as often as not, the rivot holes had been deformed from circular to elliptic shape. This led him to think about the stress magnification around the edge of elliptic holes in plates.

_{0}, Fig. 1, Inglis [1] found that the stress at the surface of, and tangential to, the elliptic hole is

### A.A. Griffith

_{1}and σ

_{2}, respectively making angles ϕ and π/2 − ϕ with the major axis of the ellipse, Fig. 2.

_{1}= 0, the Inglis solution, with the sense of increasing β reversed, gives

_{2}= 0, it gives

The values of ϕ and β for which σ_{ββ} is a maximum are found by differentiation.

Putting \( \frac{{\partial \sigma _{\beta \beta } }}{\partial \beta } = 0 \) and taking as solution \( {\text{sin 2}}\beta = A\alpha_{0} + 0(\alpha _{0}^{2} ),\) Griffith found that σ_{ββ} is a maximum at two pairs of points.

If ϕ = 0 or \( \frac{\pi }{2},\) these points are at the ends of the major and minor axes, respectively.

The two extremal values of σ_{ββ} are always of opposite sign.

_{ββ}takes its extremal values, it is necessary to evaluate

This leads to two conditions for fracture:

_{2}> σ

_{1},

- (i)
If 3σ

_{2}+ σ_{1}> 0, fracture occurs when σ_{2}>*K*, where*K*is the strength in uniaxial tension, and ϕ = 0, i.e. the fracture surface is perpendicular to σ_{2}. - (ii)If 3σ
_{2}+ σ_{1}< 0, fracture occurs when (σ_{2}− σ_{1})^{2}+ 8*K*(σ_{2}+ σ_{1})^{2}= 0 and \( \cos 2\phi = - \frac{1}{2}\frac{{\left( {\sigma _{2} - \sigma _{1} } \right)}}{{\left( {\sigma _{2} + \sigma _{1} } \right)}} \) and crack growth proceeds from near, but not at, the end(s) of the major axis of the pre-existing flaw and is on a plane inclined to the directions of principal stress as illustrated in Fig. 4.

Note that, in uniaxial compression, σ_{2} = 0, σ_{1} = −σ, equation (ii) applies and fracture is predicted when σ = −8*K*; that is the uniaxial compressive strength of an elastic solid is expected to be eight times its uniaxial tensile strength, an empirical fact well known to the ancient Greeks and Romans. Also note that, when shear is introduced, the crack rotates. In a layered material such as a laminate, this rotation enables the starter crack to seek out the inter-ply plane of weakness as illustrated in Fig. 4.

### E. Orowan

Of special interest in geophysics are the shaded areas in Fig. 5. Here, fracture is predicted when both principal stresses are compressive and strongly unequal. This non-obvious prediction has been used by Orowan to develop models for mechanical behaviour of the upper mantle; the Earth continues to cool down and radial shrinkage associated with this cooling creates circumferential shrinkage in the upper mantle which, because the mantle is a thin shell, manifests itself as two-dimensional compression in the horizontal plane.

In aeronautics, propulsion works against drag and lift works against payload to create an overall stress field that is two-dimensional. Of particular interest in materials selection for aerospace applications is the fact that the overall stress field experienced by airframes includes bidimensional compression; the possibility that the principal stresses can become strongly unequal (during “heavy landing”, for example) warrants investigation.

## Experimental bidimensional compression

## Overcoming the problem of delamination

Delamination heralds the start of mechanical failure of laminates. When subjected to repeated deformation by bending, areas of delamination appear, thereby rendering the laminate more compliant so that, for the same applied forces, it bends more, causing fibres to become sufficiently tensioned that fibre fracture commences, rendering the laminate even more compliant so that it bends even more easily and broken fibre lengths pull out. Summarising, delamination occurs first, is followed by the onset of fibre fracture, and then by fibre pull-out.

The specific fracture surface energy γ for composites based on the three-dimensional fibre weave illustrated in Fig. 8 is not expected to be isotropic. Rather, it is anticipated that γ will vary with the number of fibres intersected, i.e. γ will be orientation dependant. Over a solid angle about any chosen direction, there will be a range of orientations of planes for which the number of fibre segments intersected by unit area is constant. For example, over a solid angle of, say, 45° about a cube edge (of the mother cube) or about a face diagonal direction, it is 4, compared with 6 about body diagonal direction. For this reason, a polar diagram of γ will be similar in appearance to the surface of a raspberry in that it is constructed from spheres drawn through the origin. In the context of representation of anisotropy of surface energy of crystals, Frank [6] pointed out that a geometrically simpler representation is the γ^{−1}—polyhedron obtained by inverting these spheres through the origin when they become planes.

The weave shown in Fig. 8 constitutes a lattice. In Fig. 10, it is evident that the centres of mass (and of electron density) of this lattice are at the fibre cross-overs, suggesting that the corresponding reciprocal lattice might be that for centres of electron density located at the corners of space-filling equilateral truncated octahedra, i.e. that for simple cubic crystals. It is of some interest to consider the likely modes of wave propagation by such a periodic structure (J.C. Gill (2006) Private communication) and, in particular, the propagation of electromagnetic waves when the fibre material, but not the matrix material, is a good conductor. Laminates with fibre volume fraction η > 1/2, modelled as alternate slabs of conducting and insulating material, are a special case in that an electromagnetic wave can have **E** perpendicular to, and **B** parallel to, all conducting surfaces. This allows the boundary conditions at the surfaces to be satisfied even if the wavelength is much greater than the separation of the slabs; the wave can then propagate, without loss if the surface conductivity is infinite. However, in three-dimensional composites woven with fibres that are good conductors, the boundary conditions can be satisfied only if the directions of **E** and **B** are able to reverse in distances of the same order of magnitude as the fibre separation (d). That limits propagation to wavelengths shorter than some critical value of the order of 2d; with shorter wavelengths, the wave would propagate, again without loss, if the conductivity is infinite.

## Acknowledgements

The author gratefully acknowledges correspondence with Dr J. C. Gill, latterly of Bristol University, on the physics of electromagnetic wave propagation by fibre-reinforced materials. The mathematical equations and the figures were prepared with the assistance of T. L. Ashbee.