Other Methods of Stretching Polymer Melts

Other techniques of determining the extensional properties of polymer melts include constant-force experiments,74) such as the weight dropping extensiometer of Takaki and Bogue 73) wherein sample rods are extended isothermally at constant force. The major advantages here are that large extensions can be obtained and the mode of sample failure can be studied. Strain-time data are obtained, and these can be analyzed with the help of constitutive equations, much in the same way as with fiber-spinning data. 

A simple way of understanding the behavior of viscoelastic liquids is by analogy to mechanical models. The stress response t of a spring of modulus G connected in series to a dashpot having a damping constant t] (the combination is often called a Maxwell element) is given in one dimension by the equation 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79) 

Equation (26) is what is known as a rate-type constitutive equation, and it gives the extra stress in implicit form. Integral equations, on the other hand, are explicit in the stress, and the simplest of these can be derived based on the Boltzmann superposition principle. According to this principle, the stress in a material at any time can be obtained by adding stress tbatjndivjduallv result from earh Vio 

It can easily be shown(9) that Eq. (29) becomes identical with Eq. (24) if the stress relaxation modulus G(t-s) is chosen in the following way  

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