Shear stresses
For lap joints loaded in tension, the load is transferred predominantly by shear stresses in the adhesive layer. If the adhesive was loaded uniformly then
Where
ta is the adhesive shear stress
P is the load
L is the overlap length
b is the width
However, because the substrates are deformed in tension, the shear stresses are concentrated at the ends of the overlap, the so-called ‘shear lag’ effect as illustrated in Figure 1. When a joint is loaded, initially the adhesive is elastic, but for rigid adhesives on further loading the adhesive is stressed beyond its yield point in shear and regions of uniform stress develop at the edges of the joint. As the load is increased, these uniform shear regions will spread through the whole of the overlap and a limit will be reached when the joint can carry no further load. An upper limit on strength can therefore be derived as
Where
Pmax is the maximum joint strength possible
ty is the adhesive yield stress in shear
In practice this maximum strength is not achieved because the shear strains exceed the limits for the adhesive, the effect of peels stresses and failure of the substrate
Analytical solutions for the shear stress distribution indicate that
Where
tmax is the maximum adhesive shear stress
Ga is the shear modulus of the adhesive
Es is the Young's modulus of the substrate
ta is the thickness of the adhesive layer
ts is the thickness of the substrate
Thus the bigger the ratio between the stiffness of the substrate and the adhesive layer, the more uniform the shear stress distribution.
Peel (normal tensile) stresses
Tension stresses arise in the adhesive due to peel, cleavage or tensile loading of joints and can also occur in shear loaded joints due to bending moments that arise as a result of the eccentricity of the loading. This is illustrated for single and double lap joints in Figure 2. These stresses result in failure of the adhesive before the shear stresses are fully developed so that the theoretical maximum joint strength is not attained. Also the bending moments and tension stresses may result in yielding in metallic and thermoplastic substrates, which may also limit joint strength. The maximum peel stress like the shear stress is lower the bigger the ratio between the stiffness of the substrate and the adhesive layer
Joint strength and other properties can therefore be optimised by minimising peel stresses and maximising the stiffness ratio between substrate and adhesive layer.
A simple failure diagram shown in Figure 3 can be constructed based on the following limits:
- adhesive failure due to a fully plastic adhesive layer in shear
- substrate failure due to fully plastic section in tension and bending (metallic or thermoplastic)
In practice joint strength will fall below these limits due to the effects of adhesive peel stresses and other substrate modes of failure such as transverse tension in composites. Thus the actual failure line lies below these limits and a design limit can be set with an appropriate factor of safety.
Figure 3 also indicates that with a particular substrate thickness, joint strength will increase approximately in proportion to the overlap length for short overlaps. However the benefits diminish for long overlap lengths as the stress distribution in the adhesive becomes less sensitive to the overlap length and failure becomes more dependent on the substrate. For thicker substrates and shorter overlap lengths, joint strength will tend to depend on failure of the adhesive. For thinner substrates and longer overlap lengths, joint strength will tend to depend on failure of the substrate.
