49 Mechanics3. Mechanics of Polymers: Viscoelasticity of Wolfgang G. Knauss, Igor Emri, Hongbing Lu 3.2.6 Special Stress or Strain Histories With the heavy influx of polymers into engineering Related to Material Characterization 56 designs their special, deformation-rate-sensitive 3.2.7 Dissipation properties require particular attention. Although Under Cyclical Deformation............ 63 we often refer to them as time-dependent ma- 3.2.8 Temperature Effects...................... 63 Part A terials, their properties really do not depend on 3.2.9 The Effect of Pressure time, but time histories factor prominently in the on Viscoelastic Behavior responses of polymeric components or structures. of Rubbery Solids ......................... 68 3 Structural responses involving time-dependent 3.2.10 The Effect of Moisture and Solvents materials cannot be assessed by simply substitut- on Viscoelastic Behavior................ 69 ing time-dependent modulus functions for their elastic counterparts. The outline provided here is 3.3 Measurements and Methods.................. 69 intended to provide guidance to the experimen- 3.3.1 Laboratory Concerns ..................... 70 tally inclined researcher who is not thoroughly 3.3.2 Volumetric (Bulk) Response ........... 71 familiar with how these materials behave, but 3.3.3 The CEM Measuring System ............ 74 needs to be aware of these materials because 3.3.4 Nano/Microindentation laboratory life and applications today invariably for Measurements of Viscoelastic Properties involve their use. of Small Amounts of Material......... 76 3.3.5 Photoviscoelasticity ...................... 83 3.1 Historical Background ........................... 49 3.4 Nonlinearly Viscoelastic Material 3.1.1 The Building Blocks of the Theory Characterization ................................... 84 of Viscoelasticity .......................... 50 3.4.1 Visual Assessment of Nonlinear Behavior................... 84 3.2 Linear Viscoelasticity............................. 51 3.4.2 Characterization 3.2.1 A Simple Linear Concept: Response of Nonlinearly Viscoelastic Behavior to a Step-Function Input .............. 51 Under Biaxial Stress States ............ 85 3.2.2 Specific Constitutive Responses (Isotropic Solids) .......................... 53 3.5 Closing Remarks ................................... 89 3.2.3 Mathematical Representation of the Relaxation and Creep Functions 53 3.6 Recognizing Viscoelastic Solutions 3.2.4 General Constitutive Law for Linear if the Elastic Solution is Known.............. 90 and Isotropic Solid: Poisson Effect .. 55 3.6.1 Further Reading ........................... 90 3.2.5 Spectral and Functional Representations........................... 55 References .................................................. 92 3.1 Historical Background During the past five decades the use of polymers has signs derives in part from the ease with which these seen a tremendous rise in engineering applications. This materials can be formed into virtually any shape, and growing acceptance of a variety of polymer-based de- in part because of their generally excellent performance 50 Part A Solid Mechanics Topics in otherwise normally corrosive environments. This re- well to bear in mind that certain parts of the following cent emergence is driven by our evolving capabilities exposition are also applicable to these materials. during the last seven decades to synthesize polymers in Because the emphasis in this volume is placed on great variety and to address their processing into useful experimental methods, rather than on stress analysis shapes. methods, only a cursory review of the linearized theory Historically polymers have played a significant role of viscoelasticity is included. For the reader’s educa- in human developments, as illustrated by the intro- tional benefit a number of books and papers have been ductory comments in [3.1]. Of great consequence for listed in the Further Reading section, which can serve the survival or dominance of tribes or nations was the as resources for a more in-depth treatment. This re- development of animal-derived adhesives for the con- view of material description and analysis is thus guided struction of high-performance bows, starting with the by particular deformation histories as a background American Indian of the Northwest through the develop- for measurements addressing material characterization ments by the Tartars and leading to the extraordinary to be used in engineering design applications. Al- Part A military exploits of the Turks in the latter Middle though the nonlinearly viscoelastic characteristic of Ages [3.2]. In principle, these very old methods of these materials are not well understood in a general, producing weaponry continue to aid today in the con- three-dimensional setting, we include some reference to 3.1 struction of modern aerospace structures. While the these characteristics in the hope that this understanding current technology still uses principles exploited by our will assist the experimentalist with properly interpreting ancestors many years ago, the advent of the synthetic laboratory measurements. polymers has provided a plethora of properties avail- able for a vast range of different engineering designs. 3.1.1 The Building Blocks of the Theory This range of properties is, indeed, so large that empir- of Viscoelasticity ical methods are no longer sufficient to effect reliable engineering developments but must now be supported Forces are subject to the laws of Newtonian mechan- by optimum analytical methods to aid in the design ics, and are, accordingly, governed by the classical laws process. of motion. While relativistic effects have been stud- One characteristic of polymers is their relative sen- ied in connection with deforming solids, such concerns sitivity to load exposure for extended periods of time are suppressed in the present context. Many texts deal or to the rate of deformations imposed on them. This with Newtonian mechanics to various degrees of so- behavior is usually and widely combined under the phistication so that only a statement of the necessary concept of viscoelastic behavior, though it is some- terminology is required for the present purposes. In the times characterized as representing fading memory of interest of brevity we thus dispense with a detailed pre- the material. These time-sensitive characteristics typ- sentation of the analysis of stress and of the analysis of ically extend over many decades of the time scale strain, except for summarizing notational conventions and characteristically set polymers apart from the and defining certain variables commonly understood in normal engineering metals. While the strain-rate sen- the context of the linear theory of elasticity. We adhere sitivity [3.3] and the time dependence of failure in to the common notation of the Greek letters τ and ε de- metals [3.4] are recognized and creep as well as creep noting stress and strain, respectively. Repeated indices rupture [3.5–10] of metals is well documented, one on components imply summation; identical subscripts finds that the incorporation of rate-dependent material (e.g., τ11) denote normal components and different ones properties into models of time-dependent crack growth shear (e.g., τ12). The dilatational components of stress, – other than fatigue of intrinsically rate-insensitive ma- τii, are often written as σkk, with the strain comple- terials – still stands on a relatively weak foundation. ment being εkk. Because the viscoelastic constitutive Metallic glasses (i.e., amorphous metals) are relatively description is readily expressed in terms of deviatoric newcomers to the pool of engineering materials. Their and dilatational components, it is necessary to recall the physical properties are at the beginning of exploration, components Sij of the deviatoric stress as but it is already becoming clear through initial stud- 1 ies [3.11, 12] that their amorphous structure endows Sij = τij − τkk · δij , (3.1) them with properties many of which closely resemble 3 those of amorphous polymers. While these develop- where δij denotes the Kronecker operator. Similarly, the ments are essentially in their infancy at this time it is corresponding deviatoric strain e is written in compo- Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 51 nent form as The remaining building block of the theory consists 1 of the constitutive behavior, which differentiates vis- eij = εij − εkk · δij . (3.2) 3 coelastic materials from elastic ones. The next section For further definitions and derivations of measures is devoted to a brief definition of linearly viscoelastic of stress or strain the reader is referred to typical texts. material behavior. 3.2 Linear Viscoelasticity The framework for describing linearly viscoelastic face (boundary) of a viscoelastic solid. Specification of material behavior, as used effectively for engineer- such a quantity under uniaxial relaxation is not partic- ing applications, is phenomenological. It is based ularly useful, except to note that in the limit of short Part A mathematically on either an integral or differential for- (glassy) response its value is a limit constant, and also mulation with the material representation described under long-term conditions when the equilibrium (or realistically in numerical (tabular) or functional form(s). rubbery) modulus is effective, in which case the Pois- The fundamental equations governing the linearized son’s ratio is very close to 0.5 (incompressibility). 3.2 theory of viscoelasticity