Strength of Sandwich Structures
As we have shown in the previous section, the main advantage of sandwich core structures is that they are stiff and light. However, stiffness alone is not enough. The beam must also be strong. In considering strength there are at least 5 different modes of failure of the composite sandwich when loaded in bending; a given structure will fail at which ever mode occurs at the lowest load. The failure modes are (a) Yielding or Fracture of the tensile face, (b) Buckling or Wrinkling of the compression face, (c) Failure of the core in shear all though there is also a lesser possibility of tensile or compressive failure of the core, (d)The failure of the bond between the face and the core and (e) the possibility of indentation of the faces and core at the loading points.
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 B3 is geometric constant dependent on the mode of loading and is given in Table 1.
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 Poissons ratio for the core to be 1/3 and using the result gives
The core will fail, when the principal stresses in the core combine to exceed the yield criterion. generally, the shear stresses in the core are large compared with the normal stresses and so failure will occur when the maximum shear stress in the core exceeds the sheir yield strength of the core.This shear yield strength corresponds to plastic bending of the cell walls about the cell edges which in turn depends on the foam density and the yield strength of the faom substrate.
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* is the foam cell size. (Here we assume that the crack length a > 4l* )
Failure of the Adhesive Bond
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
If the crack in the interface has length 2a then its area is 2ab. I f that crack propagated over the entire interface (area = lb) than all the elastic strain energy (U) would be replaced. If the crack extends a distance da then the crack area changes by 2bda and the change in energy is
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.,
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/B3 the failure load is then
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.,
where A is the contact area.
Table 2 .Failure Mode equations for rectangular beams
Failure Mode||Faullire Load|
|Face Yielding|| |
|Face Wrinkling|| |
|Core Shear|| |
|Core Fracture|| |
|Bond Failure|| |
Continue with fracture maps.
H.G.Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Oxford, 1969.
S.P.Timoshenko & J.N.Goodier, Theory of Elasticity, McGraw-Hill, New York 1970, p22.
L.J.Gibson & m.F.Ashby, Cellular Solids - Structure and Properties, Pergamon Press, Oxford, 1988