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Page 1: Stress Relaxation With Finite Strain - nvlpubs.nist.gov · Stress Relaxation With Finite Strain Soon after World War II, the development and use of synthetic polymers, ... theory

Stress Relaxation With Finite Strain

Soon after World War II, the development and useof synthetic polymers, such as rubber and plastics,burgeoned. The rapidly expanding uses of these materi-als demanded a better understanding of their funda-mental properties, especially their mechanical proper-ties. At that time engineers were more accustomedto dealing with materials such as metals, whichbehaved very differently under stress than did polymers.When a reasonable load is placed on a metal, itwill bend or deform by an amount uniquely determinedby its elastic properties. Sustaining the load for a longerperiod of time does not produce any further elasticdeformation. For synthetic rubbers or polymers,such uniquely determined deformation cannot beassumed to occur. Inevitably, and often dramatically,the deformation corresponding to a given load willchange as time progresses. Such behavior is calledtime-dependent.

It was necessary, then, to learn how to deal withtime-dependent properties in these new materials, todiscover which properties were controlling and control-lable, to learn how to measure such properties, and tounderstand how to handle them in engineering design.What was needed was a rather ambitious undertaking,namely the construction of a fully three-dimensionaltheory that obeyed the laws of thermodynamics and

that went beyond the assumption commonly known aslinearity. The paper Stress Relaxation with Finite Strain[1], published in 1962 by Bernstein, Kearsley, andZapas, was a major milestone in meeting thischallenge.

By linearity we mean the following: If you double theload, you double the displacements involved in the cor-responding deformation. More generally, a model is saidto be linear if, when one multiplies a load by any factor,all displacements are multiplied by the same factor. Intheoretical models, it is more convenient to deal withrelative displacements called strains. Essentially, a strainis a change in a length divided by a reference length. Forlinearly elastic models, the strain is simply proportionalto the stress that produces it. For nonlinear models, suchsimple and unique proportionality does not occur. Thatwas the fundamental problem for models of syntheticrubber and polymers.

In early work, Herbert Leaderman in the RubberSection at NBS attempted to adapt his very successfullinear models to the new materials. He did this by tryingvarious nonlinear strain measures, with which he refor-mulated his models. But such stratagems proved to besimply perturbations, or slight variations, on the themeof linear models, and offered only very limited insightsinto the effects of nonlinearity.

Fig. 1. Louis Zapas, Elliot Kearsley, and Barry Bernstein, ca. 1991.

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Page 2: Stress Relaxation With Finite Strain - nvlpubs.nist.gov · Stress Relaxation With Finite Strain Soon after World War II, the development and use of synthetic polymers, ... theory

Beginning in 1962, a fresh attack was made on thisproblem in the Rheology Section of the MechanicsDivision of NBS by a team composed of physicist ElliotA. Kearsley and chemical engineer Louis J. Zapas of theRheology Section, together with mathematician BarryBernstein of the Mathematical Physics Section in theMathematics Division. By allowing for very wide varia-tions in characteristic times and characteristic tempera-tures, this group observed what appeared to be strikingqualitative similarities in the nonlinear viscoelasticbehavior of widely different materials. With respect tocertain characteristic quantities, the qualitative behav-iors of polymeric materials ordinarily treated as solidsand of materials ordinarily treated as liquids seemedindistinguishable. Motivated by this observation, theNBS team constructed a phenomenological model ofnonlinear viscoelasticity within the following limita-tions:

(1) It is inherently nonlinear, not a perturbation oflinear viscoelasticity.

(2) It is consistent with all the geometric and thermo-dynamic constraints appropriate to the mechan-ics of a material deforming in three-dimensionalspace.

(3) It models the known qualitative characteristics ofviscoelastic materials.

(4) It is based on a generalized micromechanicsof viscoelastic materials, but is not tied to thedetails of any particular material.

This effort resulted in a model representing a materialwhose responses in shear arise from a time-dependentpotential bearing the dimensions of entropy. Further-more, this potential depends not simply on the currentvalue of deformation, but rather on the whole past-timehistory of the deformation. In other words, to computeits value at a given time, one needs to know the historyof the deformation as a function of time. In producingelastic effects, this potential plays the role of an elasticstrain-energy function. It induces an elastic stress in acurrent configuration derived from a weighted sum ofpartial stresses arising from the deformations of pastconfigurations to the current configuration. As thecurrent configuration evolves, each particular pastconfiguration recedes in time, and the weight of itscontribution to the current stress recedes accordingly,resulting in viscous energy loss.

In addition, the notion of the “material clock” wasincorporated. Consider a temperature-sensitive clockthat runs faster at higher temperatures than at lowertemperatures. Suppose that what determines how longago in the past a particular event occurred is, to the

material, effectively the time interval on that tempera-ture-sensitive clock rather than the time interval on thetemperature-insensitive laboratory clock. Such a con-ceptual clock, with its rate adjusted appropriately for thematerial with respect to the laboratory clock, is whatone means by a material clock.

The first public presentation of this work occurred atthe annual meeting of the Society of Rheology in 1962.At that time, a version of the model suitable for useunder isothermal conditions was presented in a seriesof three talks. This presentation was followed by asummary of the model published as a paper in the 1963transactions of the Society [1]. Subsequently, a morecomplete version including thermodynamic consider-ations was published in the NBS Journal of Research in1964 [2].

For the next few years, attention to this model waspursued only at NBS and a few other laboratories whereexperimental studies were conducted to establish thesuccesses and limitations of the model for treating themechanics of polymeric materials. However, by 1970,inherent difficulties in alternative competing models ledresearchers to turn in greater numbers to the NBS

Fig. 2. Schematic of a stress-relaxation apparatus. The rigid supportcylinder determines the ratio of final to initial length (extension ratio)of the sample while the displacement of the spring measures the forceon the sample.

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Page 3: Stress Relaxation With Finite Strain - nvlpubs.nist.gov · Stress Relaxation With Finite Strain Soon after World War II, the development and use of synthetic polymers, ... theory

model. The authors had called the model “a perfectelastic fluid” by analogy to the concept of a perfect gas.However, as the number of citations to the 1963 articlebegan to climb, the model became widely known as “theBKZ fluid” or, on occasion, as “the K-BKZ fluid.”During the next decade, the use of the BKZ fluid be-came ubiquitous among rheologists. By 1988 the BKZmodel was so well established in rheological circles thatthe Society of Rheology, at its annual meeting, held asession entitled “25 Years of the BKZ Theory” to markthe 25th anniversary of its initial publication and toreview the subsequent developments. The BKZ model isnow expounded in most texts on mechanical propertiesof materials and is commonly taught in courses forstudents of polymer science and of chemical and me-chanical engineering.

According to the BKZ model, the response of a mate-rial to any deformation history can be calculated frommeasured behavior in stress-relaxation. This is a veryconvenient property for calculation. Furthermore, thelimitations of the BKZ model have been well estab-lished. Despite some failings, it has been shown to af-ford an excellent representation of the mechanical be-havior of most polymeric materials in most situations.Consequently, versions of the BKZ model are widelyused in finite-element calculations; in designing injec-tion molding and blow molding processes; in film

stretching and extrusion processes; and in designingsystems for processing and handling materials withcomplex rheology. Such processes occur, for example, inthe fabrication of plastic objects, in packaging, in foodprocessing, and in the mixing of paints and othercoatings. Modifications and ornamentation of the modelare still being proposed and tested to overcome theknown failings, and the end is not in sight. On theoccasion of the 25th anniversary of the publication ofthe BKZ paper, R. I. Tanner opined in a presentation [3],“I believe that we will continue to use the (BKZ) theory,. . . I do not think that the theory will fade away rapidlysince it is the optimal type of single-integral equation. . . . ”

Prepared by Elliot Kearsley and Barry Bernstein.

Bibliography

[1] B. Bernstein, E. A. Kearsley, and L. J. Zapas, A Study of StressRelaxation with Finite Strain, Trans. Soc. Rheol. VII, 391-410(1963).

[2] B. Bernstein, E. A. Kearsley, and L. J. Zapas, Thermodynamicsof Perfect Elastic Fluids, J. Res. Natl. Bur. Stand. 68B, 103-113(1964).

[3] R. I. Tanner, From A to BKZ in Constitutive Relations, J. Rheol.32, 673-702 (1988).

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