Rubber networks were prepared by cross-linking linear styrene-butadiene random copolymer (67% styrene content) with y-radiation. Before cross-linking, the polymer was stretched in simple extension, allowed to relax at constant strain and temperature for a controlled time, and then quenched 20 OC below Tg and maintained at this temperature during cross-linking. Upon warming above Tg, the equilibrium length was intermediate between the stretched and unstretched lengths and was describable as the result of additive contributions from the cross-link and trapped entanglement networks. The fractions of trapped unrelaxed entanglements T, and trapped relaxed entanglements Te,* were estimated and compared with the Langley theory. For minimal terminal zone relaxation before quench, T, agreed rather well with the Langley theory and Te, was zero. With increasingly long terminal zone relaxation before quench, T, increased and TeJ became finite and increased. This behavior is consistent with the terminal mechanism of relaxation in the tube theory of Doi and Edwards. These experiments confii the presence of permanently trapped entangled chain structure (or equivalent trapped topological restraints), in agreement with earlier extensive work on 1,2-polybutadiene. Uncross-linked, linear, amorphous polymers of suffi- ciently high molecular weight characteristically exhibit a zone on the time (or frequency) scale of relatively slow relaxation where the magnitude of the modulus is similar to that typical of a lightly cross-linked polymer.' This magnitude, ENo, is constant for a given undiluted polymer at fixed temperature and pressure, while the width of the plateau depends strongly on the molecular weight. To- pological constraints, traditionally referred to as entan- glements, are probably the origin of the phenomen~n;~,~ they are associated with the fact that the polymer chains, as they diffuse, can slide by one another but cannot cut across one another. The concept has been widely accepted for many years that, when permanent cross-links are in- troduced, the entangled chain structure (or some portion of it) should be trapped. Such trapped entangling should then contribute significantly to the free energy of defor- mation of the cross-linked material even at equilibrium, raising it significantly higher than could be explained from the presence of cross-links alone. Recent evidence for this view is provided by combining mechanical measurements with statistical analyses of sol-gel fractions for networks prepared both by radiation cross-linking4-' and by end- linking oligomeric chains;s it is also provided by extensive experiments on networks cross-linked in strained states,sz although the existence of trapped entangling is still dis- puted. A series of earlier papers from this laboratorysle re- ported experiments conducted upon networks of 1,2- polybutadiene cross-linked in simple extension in which the elastic effects of cross-links and of structure in the originally undeformed uncross-linked polymer were in opposition. The states of ease of such strips and their stress-strain behavior in extension past the state of ease were quantitatively describable by a two-network model where the trapped structure was identified with chain entangling in the originally uncross-linked polymer which was trapped. For 1,2-polybutadiene cross-linked in simple shear and equibiaxial extension as well, similar results have been reported by Kramer and co-w~rkers;'~ but all ex- periments on networks cross-linked in strained states have been made with this single polymer. The present paper reports further experiments on a different polymer, a styrene-butadiene random copolymer with a styrene content of 67%. The results corroborate those of the earlier papers, with the conclusion that the existence of trapped entangled chain structure is con- firmed. 0024-9297/83/2216-0039$01.50/0 Theory Polymers with entanglement and cross-link effects in opposition display elastic behavior that can be described using theories for the elastic behavior of polymers cross- linked in two stages, first in the isotropic state and then again in a strained state. The free energy of deformation of such materials has long been interpreted cla~sically~~-~~ as the sum of contributions from two independent virtual networks; the first network has as rest length the dimen- sions of the isotropic polymer, while the second has as rest length the dimensions of the polymer during the course of cross-linking in the strained state. In the experiments described below, the role of the classical first-stage cross-links is played by entanglements in the originally undeformed uncross-linked polymer which were trapped by the cross-linking. As in previous work,1@18 the cross-link network, under compression in the state of ease, is assumed to be neo- Hookean (deviations from neo-Hookean elasticity for compressed networks are known to be smallz7), but devi- ations from neo-Hookean elasticity of the entanglement network while strained in simple extension are taken into account. The phenomenological equation of Blatz, Sharda, and TschoeglZ8 (BST) is used here to describe the entan- glement network. The elastic free energy of deformation is