Hsiao-Kausch

In the fracture theory of Hsiao-Kausch [60, 61] the state of orientation of a polymeric solid is explicitly recognized. The theory combines the kinetic concept of Zhurkov or Bueche and the theory of deformation of anisotropic solids developed by Hsiao [59] 

The summation convention for repeated indices is used. E is the modulus of elasticity of the elastic elements, and s, s are components of the unit vector in the direction of orientation identified by and 0 (sj = sin cos 0, S2 = sin sin 0, S3 = cos 1 ). For uniaxial stress, 0 3 = Uq, and transverse symmetry about the 33-axis Eq. (3.24) becomes  

In a partially or nonoriented system, the microscopic axial stresses are larger than Oq by a factor of up to E/Ej where E is the longitudinal modulus of elasticity of the sub volume containing the elastic element. If all sub volumes of a sample behave identically, then E is the longitudinal modulus of the (partially) oriented sample. Naturally the largest axial stresses are experienced by those elements oriented in the direction of uniaxial stress. In a completely oriented system E and E are equal by definition, and macroscopic and microscopic stress (in the direction of orientation) are equal also. 

As stated above it is then first assumed in accordance with Zhurkov s kinetic considerations that elements break and secondly that the orientation-dependent rate of breakage of elements is given by Eq. (3.17) as 

Thirdly it is assumed that after breakage of an element mechanical equilibrium is established within the affected subvolume and that strain 33 increases. This means that the elastic elements of all orientations take over an appropriate share of the load carried previously by the broken element to balance the load acting on the 

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Fig. 3.10. Decay within a population of randomly oriented elements at high stresses (/S pQ >10 RT) essentially those elements oriented in the direction of uniaxial stress break (upper graph). TIME = linear time scale according to Eq. (3.31) (Kausch, Hsiao (601).

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