If the strip is heated using a hot air blower, the weight will rise, i. All governing equations in this theory are linear partial differential equations, which means that theprinciple of superpo. The conformations of their strands are described by the model of a combined chain, which consists of the network strand and two virtual chains attached to its ends. A good balance of theory and application for a first introduction. Basic elasticity and viscoelasticity in the physically stressful environment there are three ways in which a material can respond to external forces. In this generalization, they describe the elastomer as a set of crosslinks connected by polymer chains, which are. Equivalent freely jointed chain differs from the actual polymer on length scales of kuhn segment b or smaller, but has the same. Whereas in a phantom network devoid of such constraints the displacements of junctions from their mean positions are invariant with strain, involvements. Moreover, the presence of the fibers prevent the amorphous portion of the polymer chains from aligning themselves when subjected to a tensile force, therefore, in most cases making them brittle. This book covers in great detail the rousesegmentbased molecular theories in polymer viscoelasticity the rouse theory and the extended reptation theory based on the framework of the doiedwards theory that have been shown to explain experimental results in a consistently quantitative.
Sadler, in comprehensive polymer science and supplements, 1989. There are both internal crosslinks within a single molecule and external crosslinks between molecules. Note that when the stress is removed the strain also returns to zero. Formulation and exact solutions of some linear elasticity problems.
Figure 43 illustrates this by showing schematically the strain response to a particular stress history. The fractal theory of elasticity and rubberlike state has been derived on the base of conceptions of the scale invariant character of the polymer structure. Type of polymer modulus, e pa rubber 106 semicrystalline polymer above tg 10 8 semicrystalline polymer below tg 10 9 glassy polymer 109 semicrystalline polymer fiber 1010 polymer single crystal fiber 1011 table 1. Revisiting the elasticity theory for real gaussian phantom. When an elastic not elastomeric, or long range elastic material is stressed, there is an immediate and corresponding strain response. The strain, y, is measured in radians, and the shear modulus, g, is given by g y x. This section provides a lecture outline on polymer theory, entropy elasticity, and collapse, with detailed lecture notes and links to related materials. Above the glass transition temperature, the rubberlike polymers are in a liquidlike state and the mer or repeat units change their position readily and continuously due to brownian motion. This can be explained on the basis of rubbery elasticity theory according to which the force exerted by fixed chain ends is inversely proportional to number of statistical elements and the magnitude of end to end distance. In this book, the studies of the rouse, doiedwards, and extended reptation theories are developed in a consistent manner from a basic level and discussed in detail. This theory proved to be much more accurate than the two existing approaches to calculating polymer elasticity known as the affine network theory and phantom network model, both of which assume.
Lecture 4 part 1 rubber and polymer elasticity, polymer structure. Introduction to polymer science, chain structure and configuration, molecular weights and sizes, concentrated solutions and phase separation behavior, the amorphous state, the crystalline state, polymers in the liquid crystalline state, glassrubber transition behavior, crosslinked polymers and rubber elasticity. However, in real networks, defects such as dangling ends, cyclic defects, and polydispersity in strand sizes exist. Completely revised and updated, the fourth edition of this classic text continues to offer the reader a thorough understanding of viscoelastic behavior, essential for the proper utilization of polymers. Free polymer chemistry books download ebooks online textbooks. In particular, it fills the gap between the welldeveloped numerical methods and sophisticated methods of elasticity theory. The elasticity modulus em of each state varies greatly with the glassy state having an em anywhere between 2000 and 3000 mpa, whereas the rubbery will have an em value between 0. Unesco eolss sample chapters materials science and engineering vol. High molecular weight polymers that exhibit rubberlike behavior are known as elastomers.
Bucknall, principles of polymer engineering, 2nd edition, oxford university press, 1997. Influence of chain architecture on microdomain characteristics pdf 1. Paul flory polymer chemistry by the nobel laureate in polymer chemistry this is best known for the physical chemistry parts such as the floryhuggins theory and floryrehner elasticity theory, floryfox glass transition theory, and flory everything else about polymers theories this is the bible of polymer science. Moreover, studies have shown that cyclic defects, or loops, are intrinsic to polymer networks.
In the classical phantom network theory, the shear modulus of a polymer network is derived assuming the underlying network has a treelike topology made up of identical strands. It can add the load directly onto the forces that hold the constituent atoms or molecules together, as occurs in simple crystalline includ. The effects of restrictions on the fluctuations of network junctions imposed by neighboring chains are treated by resort to a model in which these restrictions are represented by domains of constraint. Rubber elasticity an overview sciencedirect topics. Examination of the theories of rubber elasticity using an. We used rheology, disassembly spectrometry, and simulations to measure the shear elastic modulus. Lecture 4 part 4 molecular statistics theory of polymer elasticity. Polymer network consists of long polymer chains which are crosslinked with each other to form a giant threedimensional macromolecule.
The picture that emerges from current rubber elasticity theory is one of chains frustrated by intermolecular constraints in their effort to achieve all the configurations available to an isolated chain. The important statistic for rubber elasticity is in fact the number of chains bounded by two crosslink. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. Request pdf theory of relaxation and elasticity in polymer glasses the recently developed activated barrier hopping theory of deeply supercooled polymer melts k. We evaluate the homogeneity of tetrapeg gel and examine the models predicting rubber elasticity. Also, crystallization in an already oriented polymer results in reduction retractive force2 with respect to oriented state. The viscosityradius relationship for concentrated polymer. Show that the modulus of an ideal ring is twice the modulus of a linear n2mer ii. Macknight, introduction to polymer viscoelasticity.
Like tensile strength, this is highly relevant in polymer applications involving the physical properties of polymers, such as rubber bands. Consider the gaussian statistics for the distribution of end to end distances. The theory of rubber elasticity edwards 1977 british. The nonlinear langevin equation theory of segmental relaxation, elasticity, and mechanical response of polymer glasses is extended to describe the coupled effects of physical aging, mechanical.
A body with this ability is said to behave or respond elastically. We used rheology, disassembly spectrometry, and simulations to measure the shear elastic modulus and count the numbers of topological. Imagine going to your favorite coffee shop and having the waiter inform you the pricing has changed. Explains principles, corresponding equations, and experimental methods with supporting reallife applications adds coverage of measurement techniques nanoindentation. In everyday life, highly elastic polymer materials are called rubbers. Explains principles, corresponding equations, and experimental methods with supporting reallife applications adds coverage of measurement techniques nanoindentation, atomic force microscopy. A langevinelasticitytheorybased constitutive equation for rubberlike networks and its comparison with biaxial stressstrain data. Infrared spectroscopy revealed the near absence of dangling chains. Figure 1 shows two primary polymer molecules which have formed some cross links. Equation 9 forms the basis for the classical theory of rubber elasticity. Due to their broad range of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Pdf fractal theory of elasticity and rubberlike state. Quantifying the impact of molecular defects on polymer. The book is of great interest for engineers who will find a lot of analytical formulae for very different problems covering nearly all aspects of the elastic behavior of materials.
Introduction to polymer science university of cincinnati. Lecture 4 part 3 neohookian theory and mooney rivlin. The main assumption is, that the mean square endtoend distance of a free chain in the bulk state depends on the dimensions of the surrounding chains. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as dna.
Lecture notes polymer physics materials science and. According to rivlin, 4 finite elasticity theory is almost as old as the classical elasticity theory itself. Elasticity, one of the most important properties of a soft material, is difficult to quantify in polymer networks because of the presence of topological molecular defects in these materials. Elasticity, ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed. Molecular theory of rubber elasticity polymer journal. Introduction to polymer viscoelasticity, 4th edition wiley. In a series of in uential papers, summarized in the textbook 1, warner and terentjev generalized the classical theory of rubber elasticity into the \neoclassical theory of liquid crystal elastomers. Lecture 4 part 1 rubber and polymer elasticity, polymer. Bhowmick encyclopedia of life support systems eolss 9 k g e 3k g. The properties of entangled polymer liquids are signi. Sep 15, 2016 after testing this approach on several materials, the researchers devised a theory that allows them to predict the elasticity of any polymer material. High elasticity is the most specific property of polymer materials, it is connected with the most fundamental features of ideal chains considered in the previous lecture. Introduction to polymer science, chain structure and configuration, molecular weights and sizes, concentrated solutions and phase separation behavior, the amorphous state, the crystalline state, polymers in the liquid crystalline state, glassrubber transition behavior, crosslinked polymers and rubber.
Feb 02, 2017 lecture 4 part 4 molecular statistics theory of polymer elasticity. Polymer theory 1, entropy elasticity, collapse lecture. Lecture 4 part 4 molecular statistics theory of polymer. Moreover, studies have shown that cyclic defects, or loops, are intrinsic to polymer.
Rubber elasticity has been one of the central problems in polymer science. Edwards science research council, state house, high holborn, wc1r 4ta, also at cavendish laboratory, madingley road, cambridge. Pdf fractal theory of elasticity and rubberlike state of. Youngs modulus quantifies the elasticity of the polymer. Concentration dependence of the number of elastically effective chains revealed the near absence of elastically ineffective loops and the validity of the phantom network model. The molecular theory of rubber elasticity rests on the premise, now fully validated by experiments, that alterations of the configurations of the chains comprising the network account for the. Modulus of various polymers although the above relationship for stress and strain cannot be applied to complex. Based on a reconsideration of the packing conditions in amorphous polymers, a modification of the statistical theory of rubber elasticity is derived. Viscoelastic properties of nearly monodisperse linear flexible polymers in both the entanglement and entanglementfree regions are. A related phenomenon can easily be demonstrated by hanging a weight on a strip of unfilled rubber such as that used for elastic bands.
Above the glass transition temperature, the rubberlike polymers are in a liquidlike state and the mer or repeat units change their position readily and. However, the first significant theory of large elastic deformations developed with particular reference to rubber appears to be that of mooney. With an increase in the length of polymer chains, the effects of topological entanglements become signi. The distribution of lengths of virtual chains in real. Polymer under tension ideal chain real chain f kt r2 r f 0 5 2 3 2 f kt r f r f 32 r f 1 ktb f r 0 r max linear elasticity r f nonlinear elasticity challenge problem 1. Mechanical properties of polymers encyclopedia of life.
Yeoh, in comprehensive polymer science and supplements, 1989. In a series of in uential papers, summarized in the textbook 1, warner and terentjev generalized the classical theory of rubber elasticity into the eoclassical theory of liquid crystal elastomers. Simplified polymer elasticity theory 32 0 2 3 2 2 0 2 2 r e p r r r 0 2 2 0 2 2 3 2 2 3 ln 2 3 0ln 2 2 r k t r f h k t const r r s econstk b chain chain b b b r r chain b b single polymer chains execute random walks end. The statistical theory of rubber elasticity predicts that the stress in a strained elastomer is proportional to absolute temperature.
Classical models of phantom networks suggest that the networks have a treelike structure. It is striking how a characteristic unique to polymers, i. Polymers can exist in three main physical states that include glassy, rubbery and viscous flow states. Theory of relaxation and elasticity in polymer glasses. Sep 16, 2016 elasticity, one of the most important properties of a soft material, is difficult to quantify in polymer networks because of the presence of topological molecular defects in these materials. Invoking the theory of rubber elasticity, one considers a polymer chain in a crosslinked network as an entropic spring. The condition for the loss of elasticity by polymer networks, both due to the presence of topological defects and near the gel point is discussed. A modification of rubber elasticity theory springerlink.
Classical elasticity theory an overview sciencedirect topics. Furthermore, the impact of these defects on bulk elasticity is unknown. It is defined, for small strains, as the ratio of rate of change of stress to strain. Assume local and macroscopic deformations are the same affine deformation. Calculate the income elasticity of demand and the crossprice elasticity of demand. Entropic elasticity hg hi atyticitsael forces limits of ideal chain. New theory overcomes a longstanding polymer problem mit news. The theory describes quite well the deformation behaviour of real.
These data suggest that tetrapeg gel is close to an. Free polymer chemistry books download ebooks online. After testing this approach on several materials, the researchers devised a theory that allows them to predict the elasticity of any polymer material. Classical elasticity theory an overview sciencedirect.