Nature of strain recovery in polyvinyl chloride following steady laminar shearing

Summary It is well known that viscoelastic materials, subjected to steady laminar shearing, develop a tension in the direction of shear, and it is of both practical and theoretical, interest to determine whether, this can give rise to normal components of strain recovery parallel and perpendicular to this direction, in addition to the tangential component corresponding to the applied shear. This has been investigated for the condition where there is opportunity for stress relaxation after shearing, plasticised polyvinyl chloride being used for the purpose. The hot material was sheared in a cone- and-plate rheometer at 1.0 sec^–1, and then allowed to cool without recovery being permitted. It was then removed, and suitable specimens were cut and re-heated above the shearing temperature, the resultant recovery being measured. In addition to partial shear recovery there was considerable retraction in the direction of shear. Moreover the accompanying expansion in the transverse plane was greater perpendicular to the shear planes than parallel to them. The strain recovery increased with applied shear till the latter reached 10–15, and then diminished, in agreement with previous tests on shear recovery immediately after shearing.

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Zusammenfassung Bekanntlich entwickelt viskoelastisches Material, einer stationären laminaren Scherung unterworfen, eine Spannung in Scherrichtung, und es ist von Interesse, ob bei Verschwinden der Zugspannung diese Anlaß zu Normalkomponenten der Dehnungserholung führen kann, zusätzlich zur Tangentialkomponente. Dies wurde unter Bedingungen untersucht, bei denen nach der Scherung Gelegenheit für Spannungsrelaxation gegeben war, bei plastifiziertem PC. Das heiße Material wurde in einen Konsus-Platten Rheometer bei 1.0 sec^–1 geschert und konnte dann ohne Rückfederung abkühlen. Es wurden geeignete Proben ausgeschnitten und über die Temperatur bei der Scherung hinauf aufgeheizt und anschließend die Rückfederung gemessen. Zusätzlich zu einer teilweisen Rückbildung der Scherung traten beträchtliche Schrumpfungen in der Scherrichtung auf. Außerdem war die Verdickung senkrecht zu den Scherebenen stärker als parallel. Die Deformationserholung stieg bis zu 10–15 an und verminderte sich dann in Übereinstimmung mit früheren Versuchen über Schererholung unmittelbar nach dem Scheren.

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The author wishes to record his thanks to Mr.W. C. Barry B. Sc. (Eng.) M.I.E.E., Manager of the Research Laboratories, to Mr.S. E. Goodall M. Sc. (Eng.) M. I. E. E., Chief Engineer of W. T. Henley's Telegraph Works Co., Ltd., and to the Company, for permission to publish this paper.     

Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.