Two theories of fracture of viscoelastic materials are hereunder proposed, the one based on a simple model and the other generalized thereupon, and an attempt is made to explain the dependence of stress and strain at break on temperature and strain rate as particularly was called the failure envelope by T.L. Smith.
The model of the first simple theory consists of two Maxwell elements (system 1 and 2) connected in parallel and the following criteria for fracture are introduced.
(1) Fracture occurs first at the system 1, and then at the system 2 where the whole load is applied.
(2) Fracture of the system 1 occurs either when the spring reaches the critical strain ε11c (in the case of large strain rate) or the dashpot does so to ε12c (in the case of small strain rate).
For the deformation of constant rate R, the following results are obtained, which explain the experimental behaviors well at least qualitatively.
at larger strain rates
at smaller strain rates
where σ, ε, G and τ follow the ordinary use and suffices 1 and 2 mean system 1 and 2 respectively and the suffix b does so "at break".
Next the above model theory is so extended to the generalized Maxwell bodies as to read that the stress of deformation at constant rate is expressed by the equation
In this case the storage energy Wst and the dissipation energy Wdis of deformation are calculated after Landel, and the following criterion is introduced, that is, the sample breaks either when the elastic part with its own modulus G0 (the instantaneous modulus) reaches the critical strain ε1c or the viscous part with its steady flow viscosity η0 reaches the critical strain ε2c.
The results are given as
at larger strain rates
at smaller strain rates
where G' and η' are dynamic modulus and viscosity respectively. Considering the dependence of G' and η' on shear rate and temperature, the failure envelope can be explained with these equations.
