Fracture mechanics is concerned with defining limits in joint performance from considerations of crack growth through an adhesive layer. It is therefore relevant to fracture, fatigue and creep resistance. As polymeric materials, various processes around the crack tip of adhesive materials can affect their resistance to crack growth.
- shear yielding, usually under compression loads
- micro-void formation or crazing, usually under hydrostatic tension initiating from inhomogeneities in the microstructure
- localised cavitation at particle-matrix interfaces, particularly relevant for toughened adhesives
- viscoelastic losses in flexible adhesives
Two approaches have been used to quantify crack growth behaviour: (i) the stress intensity factor approach, concerned with crack tip stresses and (ii) the strain energy release rate approach, concerned with the energy required to create new crack surfaces.
Stress intensity factor
For elastic materials, the stress field around a crack tip is singular. The stress intensity factor, K, represents the strength of the singularity. The general form of the stress intensity equation for a material is:
so is the applied stress, 2a is the crack length, r and q are the co-ordinates of a point and sij are the components of the stress tensor at that point. By combining Equations 1 and 2, the stress intensity can be expressed as:
Where
is the applied stress.
2a is the crack length
r and 
are the co-ordinates of a point
are the components of the stress tensor at that point
By combining Equations 1 and 2, the stress intensity can be expressed as:
where Q is a geometry factor.
The stress intensity approach then compares the K values with a material property called the fracture toughness, Kc, which is the critical value for fracture in the material. Some reservations have been expressed about the stress intensity approach. This is because it has been suggested that a thin adhesive layer would cause a complex stress field ahead of the crack, that would not be of the conventional r1/2 singularity form as in Equation 2 and which would invalidate the analysis.
Strain energy release rate
A second concept is the energy release rate and this relates the formation of new fracture surfaces to the energy released from a stress field. For a uniformly loaded joint, the energy release rate, G, can be derived from energy balance considerations as
where P is the load, b is the width of the joint, C is the compliance of the sample and a the crack length.
From appropriate tests, a critical strain energy release rate, Gc, for fracture can be derived or the relationship between the rate of crack growth and G can be derived for crack growth during creep or cyclic fatigue. These are then properties of the adhesive system. For rigid adhesives, these properties will tend to depend on the mode of loading of the joint as indicated in diagrams shown below. Mode I consists of tension across the crack leading to crack opening; mode II is a shearing action and mode III a tearing action. The weakest mode is considered to be Mode I, although combinations of the modes are the most likely situations to exist in real joints.
Fracture mechanics based joint analysis
Joint fatigue life can be predicted using linear-elastic fracture mechanics. Strain energy release rate (G) is derived as a function of crack length from the fracture analysis, usually from a finite element model of the structure or joint. This together with a fatigue crack growth model for the adhesive system can be used to predict crack growth and hence fatigue life for the range of loads of interest. The methodology is being developed so that it can be used as a tool for computer aided engineering (CAE). This approach is inherently conservative, in that any additional contributions to fatigue life due to crack initiation processes are neglected.
Figure 2 Variety of test piece geometries used in fracture mechanics testing.
Various test pieces have been developed for measurement of joint properties, generally using beam type geometries, some of which as illustrated in Figure 2. The tapered double cantilever beam was developed to give constant G with crack length for constant load. Parallel beams are easier to manufacture and can be loaded in a variety of ways to achieve various modes of loading.
Mode I (peel) is regarded as the most severe form of loading, when compared to mode II (shear) loading. Fracture toughness and fatigue crack growth resistance under mode I loading are typically an order of magnitude lower than those measured under mode II loading. A mixture of modes I and II would be expected to fall somewhere between the two. However for some adhesive systems, a mixture of mode I and mode II results in the locus of failure being closer to the interface than under mode I loading, resulting in a lower resistance to mixed mode loading.
Figure 3 Schematic fatigue resistance plots showing the possible effects of different test environments.
Fatigue crack growth data for a toughened epoxy is shown in Figure 3. For design it is important that fatigue loads do not exceed the threshold value (Gth), or a value below it, to include a safety margin.
