Before examining the specifics of each failure mode we need to establish the distribution of stress in the faces and core as a result of bending, these are and for the normal stresses in the face and core and and for the shear stresses. The maximum normal stresses are related to the bending moment, M and the distance from the centerline, y.

where B

The shear stresses vary parabolically through the thickness of the face and core, but if the faces are much thinner and stiffer than the core (as, indeed, they always are) then the shear stress can be treated as linear through the face and constant in the core

where Q , is the maximum shear force in the beam and , is the average shear stress in the face (equal to half the maximum shear stress). Finally, given the shear and normal stresses we can evaluate the principal stresses and the maximum shear stresses in the face and core (Timoshenko & Goodier,1970)

where s and t are the normal and shear stresses in either the face or the core. In the faces, the ratio of , is small so ; while in the core the ratio is generally, but not always, large because of the low core modulus. In this case

Face yielding occurs when the normal tensile stress in the face equals the strength (yield strength for ductile face materials, fracture strength for brittle face materials) so

Face wrinkling occurs when the normal stress in the compression face of the beam reaches the level of instability (Allen, 1969)

While this looks complex it can be simplified by assuming Poisson’s ratio for the core to be 1/3 and using the result gives

However, if the core is brittle, then we need to recognise that a crack like defect can initiate failue when the maximum principal stress exceeds a critical value that depends on both the fracture toughness of the core material and the crack length, i.e.

where l

Failure of the adhesive bond between the face and the core is a very important failure mechanism and the most difficult to analyse. For PolymerMatrix Composite skins bonded to foam cores, the epoxy used to make the bond is usually stronger than the foam and failure will generally occur in the foam, not the bond. However, if a crack like defect exists in the bond - such as might arise from incomplete coverage of the adhesive or a trapped air pocket for in-situ foamed composites, then failure can occur in the bond plane by extension of the crack. Faulire is analysed in terms of energy. The energy stored in the beam is

where q is the angle of bend for a given bending moment M. Using then

You should recall from our analysis if fracture mechanics that the strain energy release rate, G, is defined as the rate of release of energy per unit are crack extension, i.e.,

giving

If the rate of strain energy release exceeds the critical strain energy release rate for the adhesive then the crack will propagate and the beam willfail, Using M=Pl/B

where G is the strain energy release rate of the adhesive (or core if this is lower).

The problem of indentation of the core only occurs when loads are highly localised and arises because the stress immediately under the loading point (P/A) is gretaer than the crushing strength of the core; i.e.,

Table 2 .Failure Mode equations for rectangular beams

Failure Mode | Faullire Load |

Face Yielding | |

Face Wrinkling | |

Core Shear | |

Core Fracture | |

Bond Failure |

S.P.Timoshenko & J.N.Goodier,

L.J.Gibson & m.F.Ashby,