One can easily solve the differential equations for viscoelastic materials
∂ε/∂t=σ0/η and cρ∂θ/∂t=ασ0/Jη∂ε/∂t+k∂2θ/∂x2
simultaneously with the initial condition of the temperature rise
δ(x)=q0/2π∞∫-∞exp(iωx)dω
in such the simplest case that σ0 is a constant tensile stress and 1/η is approximated by a linear function of temperature increase, θ. The first term in the right-hand side of the second equation, representing the heat source, is derived from the work done by the stress per unit of volume per unit time. α is a factor less than (but differing little from) unity and J the mechanical equivalent of heat.
Our object in solving the equations is this: linear polymer may increase in temperature by some chance in the neighbourhood of a certain section of the material before break and the coefficient of viscosity may decrease there. It facilitates the subsequent extension of the same part, and finally leads to melting and then to breaking. We have good reason for the existence of the pattern of such an origin among several fracture patterns appearing on the fracture surface.
Thus, the fracture energy is expected to be estimated as the work done on the fracture surface layer of finite depth subjected to large local flow, and obtained as
WB=Jcρ/αxBθB,
where xB and θB are the thickness of fracture surface and the mean temperature rise there respectively.
This approach is regarded as worthy of further consideration, because of the identity of the order of magnitude of the calculated energy with those of the current data and because of the deduced reasonable relationships between the depth of the fracture surface and the thermal properties of the materials, the temperature dependency of the coefficient of viscosity and the acting stress.
