Analytical solutions for beam specimens used in fracture-mechanics testing of composites and adhesively-bonded joints typically use a beam on an elastic foundation model which assumes that a non-infinite, linear-elastic stiffness exists for the beam on the elastic foundation in the region ahead of the crack tip. Such an approach therefore assumes an elastic-stiffness model but without the need to assume a critical, limiting value of the stress,σ max, for the crack tip region. Hence, they yield asingle fracture parameter, namely the fracture energy,G c. However, the corresponding value ofσ max that results can, of course, be calculated from knowledge of the value ofG c. On the other hand, fracture models and criteria have been developed which are based on the approach thattwo parameters exist to describe the fracture process: namelyG candσ max. Hereσ max is assumed to be a critical,limiting maximum value of the stress in the damage zone ahead of the crack and is often assumed to have some physical significance. A general representation of the two-parameter failure criteria approach is that of the cohesive zone model (CZM). In the present paper, the two-parameter CZM approach has been coupled mainly with finite-element analysis (FEA) methods. The main aims of the present work are to explore whether the value ofσ max has a unique value for a given problem and whether any physical significance can be ascribed to this parameter. In some instances, both FEA and analytical methods are used to provide a useful crosscheck of the two different approaches and the two different analysis methods.