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 A Part 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- atA Part 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 atA Part 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 are the same as those for the linearized theory of elasticity, except that the 3.2.1 A Simple Linear Concept: generalized Hooke’s law of elasticity is replaced by Response to a Step-Function Input a constitutive description that is sensitive to the mater- ial’s (past) history of loading or deformation. It will be It is convenient for instructional purposes to consider the purpose of the immediately subsequent subsections that the stress can be described, so that the strain fol- to summarize this formalism of material description in lows from the stress. The reverse may hold with equal preparation for various forms of material characteriza- validity. In general, of course, neither may be prescribed tion. Little or no reference is made to general solution a priori, and a general connection relates them. The methods for viscoelastic boundary value problems. For structure of the linear theory must be completely sym- this purpose the reader is referred to the few texts avail- metric in the sense that the mathematical formulation able as listed in Sect. 3.6.1. applies to these relations regardless of which variable is Rather than repeating the theory as already outlined considered the prescribed or the derived one. For intro- closely in [3.13] we summarize below the concepts ductory purposes we shall use, therefore, the concept of and equations most necessary for experimental work; if a cause c(t) (input) and an effect e(t) (output) that are necessary, the reader may consult the initially cited ref- connected by a functional relationship. The latter must erence(s) (Sect. 3.6.1) for a more expansive treatment. be linear with respect to (a) the amplitude (additivity In brief, the viscoelastic material functions of first-order with respect to magnitude) and (b) time in the sense that interest are given in Table 3.1. they obey additivity independent of time. Note the absence of a generic viscoelastic Poisson It is primarily a matter of convenience that the function, because that particular response is a functional cause-and-effect relation is typically expressed with the of the deformation or stress history applied to the sur- aid of a step-function cause. Other representations are
Table 3.1 Nomenclature for viscoelastic material functions Type of loading Shear Bulk Uniaxial Mode extension Relaxation μ(t) K(t) E(t) Quasistatic Creep J(t) M(t) D(t) Storage μ (ω) K (ω) E (ω) Strain prescribed Loss μ (ω) K (ω) E (ω) Harmonic Storage J (ω) M (ω) D (ω) Stress prescribed Loss J (ω) M (ω) D (ω) 52 Part A Solid Mechanics Topics
feasible and we shall address a common one (steady- Condition (b) entails then that, if two causes state harmonic) later on as a special case. For now, c1(t1) ≡ α1 · h(t1)andc2(t2) ≡ α2 · h(t2), imposed at let E(t, t1) represent a time-dependent effect that re- different times t1 and, t2 act jointly, then their corres- sults from a step cause c(t1) = h(t1)ofunit amplitude ponding effects α1 · E(t, t1)andα2 · E(t, t2) is their sum imposed at time t1; h(t1) denotes the Heaviside step while observing their proper time sequence. Let the function applied at time t1. common time scale start at t = 0; then the combined For the present we are concerned only with non- effect, say e(t), is expressed by aging materials, i. e. with materials, the intrinsic prop- ≡ + → = α · − erties of which do not change with time. (With this c(t) c1(t1) c2(t2) e(t) 1 E(t t1) definition in mind it is clear that the nomenclature time- + α1 · E(t − t2) . dependent materials in place of viscoelastic materials is (3.3) really a misnomer; but that terminology is widely used, nevertheless.) We can assert then that for a non-aging Specifically, here the first response does not start until atA Part material any linearity of operation, or relation between the time t1 is reached, and the response due to the sec- an effect and its cause, requires satisfaction of ond cause is not experienced until time t2, as illustrated in Fig. 3.1. 3.2 Postulate (a): proportionality with respect to amplitude, Having established the addition process for two and causes and their responses, the extension to an arbitrary Postulate (b): additivity of effects independent of the number of discrete step causes is clearly recognized as time sequence, when the corresponding a corresponding sum for the collective effects e(tn), up causes are added, regardless of the re- to time t, in the generalized form of (3.3), namely spective application times. e(t) = αn E(t − tn) ; (tn < t) . (3.4)
Condition (a) states that, if the cause c(0) elic- This result may be further generalized for causes , its E(t 0), then a cause of different amplitude, say represented by a continuous cause function of time, ≡ α · α c1(0) h(0), with a constant, elicits a response say c(t). To this end consider a continuously vary- α · , E(t 0). Under the non-aging restriction this relation ing function c(t) decomposed into an initially discrete is to be independent of the time when the cause starts approximation of steps of finite (small) amplitudes. ≡ α · → α , > to act, so that c1(t1) h(t1) E(t t1); t t1 also With the intent of ultimately proceeding to the limit of holds. This means simply that the response–effect re- infinitesimal steps, note that the amplitude of an indi- lation shown in the upper part of Fig. 3.1 holds also vidual step amplitude at, say, time τn is given by for a different time t2, which occurs later in time than t . 1 αn → Δc(τn) = (Δc/Δτ) Δτ, (3.5) τn
Stress Strain which, when substituted into (3.4), leads to e(t) = E(t − tn)(Δc/Δτ) Δτ. (3.6) τn In the limit n →∞(Δτ → dτ), the sum Δc t1 Time t1 Time e(t) = lim E(t − tn) Δτ, (3.7) Δτ τn Stress Strain passes over into the integral 2 2 t dc(τ) 1 1 e(t) = E(t − τ) dτ. (3.8) dτ 0 t t t t 1 2 Time 1 2 Time Inasmuch as this expression can contain the effect of Fig. 3.1 Additivity of prescribed stress steps and corres- a step-function contribution at zero time of magnitude ponding addition of responses c(0), this fact can be expressed explicitly through the Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 53 alternate notation effect, one obtains the inverse relation(s) t t dc(τ) τ e(t) = c(0)E(t) + E(t − τ) dτ, (3.9) 1 d τ ε(t) = J(t − ξ) dξ (3.12) d 2 dξ 0+ 0 + where the lower integral limit 0 merely indicates that t 1 1 dτ the integration starts at infinitesimally positive time so = τ(0)J(t) + J(t − ξ) dξ, (3.13) as to exclude the discontinuity at zero. Alternatively, the 2 2 dξ + same result follows from observing that for a step dis- 0 continuity in c(t) the derivative in (3.5) is represented by where now the function E ≡ J(t) is called the shear the Dirac delta function δ(t). In fact, this latter remark creep compliance, which represents the creep response holds for any jump discontinuity in c(t) at any time, af- of the material in shear under application of a step shear ter and including any at t = 0. In mathematical terms stress of unit magnitude as the cause. this form is recognized as a convolution integral, which A Part in the context of the dynamic (vibration) response of Bulk or Dilatation Response linear systems is also known as the Duhamel integral. Let εii(t) represent the first strain invariant and σ jj(t)the corresponding stress invariant. The latter is recognized 3.2 3.2.2 Specific Constitutive Responses as three times the pressure P(t), i. e., σ jj(t) ≡ 3P(t). In (Isotropic Solids) completely analogous fashion to (3.12)and(3.13)the bulk behavior, governed by the bulk relaxation modulus For illustrative purposes and to keep the discussion K(t) ≡ E(t), is represented by within limits, the following considerations are limited to isotropic materials. Recalling that the stress and strain t dεii(ξ) states may be decomposed into shear and dilatational σ jj = 3 K(t − ξ) dξ (3.14) dξ contributions (deviatoric and dilatational components), 0 we deal first with the shear response followed by the t volumetric part. Thermal characterization will then be dεii(ξ) = 3εii(0)K(t) + 3 K(t − ξ) dξii . dealt with subsequently. dξ 0+ Shear Response (3.15) τ ε Let denote any shear stress component and its Similarly, one writes the inverse relation as corresponding shear strain. Consider ε to be the cause and τ its effect. Denote the material characteristic E(t) t 1 dσ jj(ξ) for unit step excitation from Sect. 3.2.1 in the present ε = M t − ξ ξ (3.16) ii ( ) ξ d shear context by μ(t). This function will be henceforth 3 d 0 identified as the relaxation modulus in shear (for an t isotropic material). It follows then from (3.8)and(3.9) 1 1 dσ jj(ξ) = σ jj(0)M(t) + M(t − ξ) dξii , that 3 3 dξ t 0+ dε(ξ) (3.17) τ t = μ t − ξ ξ (3.10) ( ) 2 ( ) ξ d d ≡ 0 where the function M(t) E(t) represents now the di- t latational creep compliance (or bulk creep compliance); dε(ξ) = 2ε(0)μ(t) + 2 μ(t − ξ) dξ. (3.11) in physical terms, this is the time-dependent fractional dξ volume change resulting from the imposition of a unit + 0 step pressure. The factor of 2 in the shear response is consistent with elasticity theory, inasmuch as in the limits of short- and 3.2.3 Mathematical Representation long-term behavior all viscoelasticity relations must re- of the Relaxation and Creep Functions vert to the elastic counterparts. If one interchanges the cause and effect by letting Various mathematical forms have been suggested and the shear stress represent the cause, and the strain the used to represent the material property functions 54 Part A Solid Mechanics Topics
analytically. Preferred forms have evolved, with pre- where ξ = ηm/μm is called the (single) relaxation time. cision being balanced against ease of mathematical Similarly, applying a step stress (force) of magnitude use or a minimum number of parameters required. to the Voigt element engenders a time-dependent sepa- All viscoelastic material functions possess the com- ration (strain) of the force-application points described mon characteristic that they vary monotonically with by time: relaxation functions decreasing and creep func- τ0 ε(t) = [1 − exp(−t/ς)] , (3.19) tions increasing monotonically. A second characteristic μν of realistic material behavior is that time is (almost) where ς = ην/μν is now called the retardation time invariably measured in terms of (base 10) logarithmic since it governs the rate of retarded or delayed motion. units of time. Thus changes in viscoelastic response Note that this representation is used for illustration pur- may appear to be minor when considered as a func- poses here and that the retardation time for the Voigt tion of the real time, but substantial if viewed against material is not necessarily meant to be equal to the a logarithmic time scale. atA Part relaxation time of the Maxwell solid. It can also be Early representations of viscoelastic responses were easily shown that this is not true for a standard lin- closely allied with (simple) mechanical analog mod- ear solid either. By inductive reasoning, that statement els (Kelvin, Voigt) or their derivatives. Without delving 3.2 holds for arbitrarily complex analog models. The re- into the details of this evolutionary process, their gen- laxation modulus and creep compliance commensurate eralization to broader time frames led to the spectral with (3.18) (Maxwell model) and (3.19) (Voigt model) representation of viscoelastic properties, so that it is for the Wiechert and Kelvin models (Fig. 3.2c,d) are, useful to present only the rudiments of that devel- respectively opment. The building blocks of the analog models are the Maxwell and the Voigt models illustrated in μ(t) = μ∞ + μn exp(−t/ξn) (3.20) Fig. 3.2a,b. In this modeling a mechanical force F n corresponds to the shear stress τ and, similarly, a dis- and placement/deflection δ corresponds to a strain ε. Under J(t) = Jg + Jn[1 − exp(−t/ςn)]+η0t , (3.21) a stepwise applied deformation of magnitude ε0 –sep- arating the force-application points in the Maxwell n η η → model – the stress (force) abates or relaxes by the where Jg and 0 arise from letting 1 0(thefirst μ → relation Voigt element degenerates to a spring) and n 0(the last Voigt element degenerates to a dashpot). These se- ries representations with exponentials are often referred τ(t) = ε0μm exp(−t/ξ) , (3.18) to as Prony series. As the number of relaxation times increases indefi- a) F b) c) F F nitely, the generalization of the expression for the shear relaxation modulus, becomes μ ∞ μ ξ 1 η1 d μη μ = μ∞ + ξ −t/ξ , (3.22) (t) H( ) exp( ) ξ η μ η 0 2 2 ξ F F where the function H( ) is called the distribution func- tion of the relaxation times, or relaxation spectrum, for μ η 3 3 short; the creep counterpart presents itself with the help d) F of the retardation spectrum L(ζ)as ∞ ζ μ∞ μ1 μ2 μ3 μj d μ J(t) = Jg + L(ζ)[1 − exp(−t/ζ)] + ηt , j ηj ζ η η η η 1 2 3 j 0 (3.23) F F Note that although the relaxation times ξ and the retar- Fig. 3.2a–d Mechanical analogue models: (a) Maxwell, dation times ζ do not, strictly speaking, extend over the (b) Voigt, (c) Wiechert, and (d) Kelvin range from zero to infinity, the integration limits are so Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 55 assigned for convenience since the functions H and L 3.2.5 Spectral and Functional can always be chosen to be zero in the corresponding Representations part of the infinite range. A discrete relaxation spectrum in the form 3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect H(ξ) = μnξnδ(ξn) , (3.25)
One combines the shear and bulk behavior exempli- where δ(ξn) represents the Dirac delta function, clearly fied in (3.13), (3.16)and(3.19), (3.20) into the general leads to the series representation (3.20) and can trace stress–strain relation the modulus function arbitrarily well by choosing the t number of terms in the series to be sufficiently large; ∂ε 2 kk a choice of numbers of terms equal to or larger σij(t) = δij K(t − τ) − μ(t − τ) dτ 3 ∂τ than twice the number of decades of the transition − 0 is often desirable. For a history of procedures to A Part t determine the coefficients μ see the works by Hop- ∂εij n + μ t − τ τ, (3.24) 2 ( ) ∂τ d kins and Hamming [3.17], Schapery [3.18], Clauser 3.2 0− and Knauss [3.19], Hedstrom et al. [3.20], Emri and Tschoegl [3.21–26], and Emri et al. [3.27, 28], all of where δij is again the Kronecker delta. which battle the ill-conditioned nature of the numerical Poisson Contraction determination process. This fact may result in physi- A recurring and important parameter in linear elas- cally inadmissible, negative values (energy generation), ticity is Poisson’s ratio. It characterizes the contrac- though the overall response function may be rendered tion/expansion behavior of the solid in a uniaxial stress very well. A more recent development that largely state, and is an almost essential parameter for deriving circumvents such problems, is based on the trust re- other material constants such as the Young’s, shear or gion concept [3.29], which has been incorporated into bulk modulus from each other. For viscoelastic solids MATLAB, thus providing a relatively fast and read- the equivalent behavior cannot in general be charac- ily available procedure. The numerical determination of terized by a constant; instead, the material equivalent these coefficients occurs through an ill-conditioned in- to the elastic Poisson’s ratio is also a time-dependent tegral or matrix and is not free of potentially large errors function, which is a functional of the stress (strain) in the coefficients, including physically inadmissible history imposed on a uniaxially stressed material sam- negative values, though the overall response function ple. This time-dependent function covers basically the may be rendered very well. same (long) time scale as the other viscoelastic re- Although expressions as given in (3.22)and(3.23) sponses and is typically measured in terms of 10–20 render complete descriptions of the relaxation or creep ξ ξ decades of time at any one temperature. However, com- behavior once H( )orL( ) are determined for any pared to these other functions, its value changes usually material in general, simple approximate representations from a maximum value of 0.35 or 0.4 at the short can fulfill a useful purpose. Thus, the special function . n end of the time spectrum to 0 5 for the long time μ0 − μ∞ ξ0 frame. Several approximations are useful. In the near- H(ξ) = exp(−ξ0/ξ) (3.26) Γ (n) ξ glassy domain (short times) its value can be taken as a constant equal to that derived from measurements with the four parameters μ0, μ∞, ξ0,andn representing well below the glass-transition temperature. In the long material constants, where Γ (n) is the gamma function, time range for essentially rubber-like behavior the ap- leads to the power-law representation for the relaxation proximation of 0.5 is appropriate, though not if one response wishes to convert shear or Young’s data to bulk be- μ0 − μ∞ havior, in which case small deviations from this value μ(t) = μ∞ + . (3.27) (1 + t/ξ )n can play a very significant role. If knowledge in the 0 range between the (near-)glassy and (near-)rubbery do- This equation is represented in Fig. 3.3 for the par- 2 5 −4 main are required, neither of the two limit constants ameter values μ∞ = 10 , μ0 = 10 , ξ0 = 10 ,and are strictly appropriate and careful measurements are n = 0.35. It follows quickly from (3.22) and the fig- required [3.14–16]. ure that μ0 represents the modulus as t → 0, and μ∞ 56 Part A Solid Mechanics Topics
3.2.6 Special Stress or Strain Histories log modulus 6 Related to Material Characterization For the purposes of measuring viscoelastic properties in 5 the laboratory we consider several examples in terms of shear states of stress and strain. Extensional or compres- 4 sion properties follow totally analogous descriptions.
3 Unidimensional Stress State We call a stress or strain state unidimensional when it 2 involves only one controlled or primary displacement or stress component, as in pure shear or unidirectional extension/compression. Typical engineering character- atA Part 1 –10 –5 0 5 10 izations of materials occur by means of uniaxial log t (tension) tests. We insert here a cautionary note with Fig. 3.3 Example of the power-law representation of a re- respect to laboratory practices. In contrast to working 3.2 laxation modulus with metallic specimens, clamping polymers typically introduces complications that are not necessarily to- its behavior as μ(t →∞); ξ0 locates the central part tally resolvable in terms of linear viscoelasticity. For of the transition region and n the (negative) slope. It example, clamping a tensile specimen in a standard bears pointing out that, while this functional represen- test machine with serrated compression claps intro- tation conveys the generally observed behavior of the duces a nonlinear material response such that, during relaxation phenomenon, it usually serves only in an ap- the course of a test, relaxation or creep may occur un- proximate manner: the short- and long-term modulus der the clamps. Sometimes an effort is made to alleviate limits along with the position along the log-time axis this problem by gluing metal tabs to the end of spec- and the slope in the mid-section can be readily adjusted imens, only to introduce the potential of the glue line through the four material parameters, but it is usually to contribute to the overall relaxation or deformation. a matter of luck (and rarely possible) to also represent If the contribution of the glue line to the deformation the proper curvature in the transitions from short- and is judged to be small, an estimate of its effect may be long-term behavior. Nevertheless, functions of the type derived with the help of linear viscoelasticity, and this (3.26)or(3.27) can be very useful in capturing the es- should be stated in reporting the data. sential features of a problem. With respect to fracture For rate-insensitive materials the pertinent property Schapery draws heavily on the simplified power-law is Young’s modulus E. For viscoelastic solids this con- representation. stant is supplanted by the uniaxial relaxation modulus An alternative representation of one-dimensional E(t) and its inverse, the uniaxial creep compliance D(t). viscoelastic behavior (shear or extension), though not Although the general constitutive relation (3.24) can accessed through a distribution function of the type be written for the uniaxial stress state (σ11(t) = σ0(t), described above, is the so-called stretch exponential say, σ22 = σ33 = 0), the resulting relation for the uni- formulation; it is often used in the polymer physics axial stress is an integral equation for the stress or community and was introduced for torsional relaxation strain ε11(t), involving the relaxation moduli in shear by Kohlrausch [3.30] and reintroduced for dielectric and dilatation. In view of the difficulties associated with studies by Williams and Watts [3.31]. It is, there- determining the bulk response, it is not customary to fol- fore, often referred to as the KWW representation. low this interconversion path, but to work directly with In the case of relaxation behavior it takes the form the uniaxial relaxation modulus E(t) and/or its inverse, (with the addition of the long-term equilibrium modu- the uniaxial creep compliance D(t). Thus, if σ11(t)is μ lus ∞), the uniaxial stress and ε11(t) the corresponding strain, one writes, similar to (3.10)and(3.12), μ = μ + μ − /ξ β . (t) ∞ 0 exp (t 0) (3.28) t dε (ξ) σ (t) = ε (0)E(t) + E(t − ξ) 11 dξ Further observations and references relating to this 11 11 dξ representation are delineated in [3.13]. 0+ Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 57
ε ε ε Fig. 3.4 Superposition of linear func- tions to generate a ramp
ε0 =+
t0 t t0 t t0 t
and the inverse relation as modulus (3.22) together with the convolution relation t (3.10) to render, with ε˙(t) = const = ε˙ and ε(0) = 0, the σ ξ 0 d 11( ) general result ε (t) = σ (0)D(t) + D(t − ξ) dξ. A Part 11 11 dξ 0+ t We insert here a cautionary note with respect to τ(t) = 2 μ(t − ξ)ε˙0 dξ 3.2 laboratory practices: In contrast to working with metal- 0 lic specimens, clamping polymers typically introduces t ∞ − ξ ς complications that are not necessarily totally resolvable t d = 2ε˙0 μ∞ + H(ς)exp − du in terms of linear viscoelasticity. For example, clamping ς ς a tensile specimen in a standard test machine with ser- 0 0 rated compression clamps introduces nonlinear material (3.29) t t response such that during the course of a test relax- 1 ation or creep may occur under the clamps. Sometimes = 2με˙ μ(u)du = 2ε˙ t · μ(u)du 0 0 t an effort is made to alleviate this problem by gluing 0 0 metal tabs to the end of specimens, only to introduce = 2ε(t)μ¯ (t) ; t − ξ ≡ u . (3.30) the potential of the glue-line to contribute to the over- all relaxation or deformation. If the contribution of the μ¯ = 1 t μ Here (t) t 0 (u)du is recognized as the relaxation glue-line to the deformation is judged to be small an es- modulus averaged over the past time (the time-averaged timate of its effect may be derived with the help of linear relaxation modulus). viscoelasticity, and such should be stated in reporting the data. Ramp Strain History. A recurring question in viscoelas- tic material characterization arises when step functions Constant-Strain-Rate History. A common test method are called for analytically but cannot be supplied ex- for material characterization involves the prescription of perimentally because equipment response is too slow a constant deformation rate such that the strain increases or dynamic (inertial) equipment vibrations disturb the linearly with time (small deformations). Without loss input signal: In such situations one needs to determine of generality we make use of a shear strain history in the error if the response to a ramp history is supplied the form ε(t) = ε˙ t (≡ 0fort ≤ 0, ε˙ = const for t ≥ 0) 0 0 instead of a step function with the ramp time being t0. and employ the general representation for the relaxation To provide an answer, take explicit recourse to pos- tulate (b) in Sect. 3.2.1 in connection with (3.29)/(3.30) to evaluate (additively) the latter for the strain histories ε τ µ(t) Error showninFig.3.4. To arrive at an approximate result as a quantitative guide, let us use the power-law represen-
ε0 tation (3.27) for the relaxation modulus. Making use of Taylor series approximations of the resulting functions for t 0 one arrives at (the derivation is lengthy though straightforward) t0 t t0 t Fig. 3.5 Difference in relaxation response resulting from τ(t) n t0/ξ0 = μ(t) 1 + + ... (3.31) step and ramp strain history 2ε0 2 (1 + t/ξ0) 58 Part A Solid Mechanics Topics
as long as μ∞ can be neglected relative to μ0 (usually the kernel (material) functions from modulus or creep on the order of 100–1000 times smaller). The deriva- data involving Volterra equations of the first kind tion is lengthy though straightforward. The expression can lead to sizeable errors, whether the functions in the square brackets contains the time-dependent error are sought in closed form or chosen in spectral or by which the ramp response differs from the ideal relax- discrete (Prony series) form [3.18, 19, 27]. On the ation modulus, as illustrated in Fig. 3.5, which tends to other hand, Volterra equations of the second kind do zero as time grows without limit beyond t0. not suffer from this mathematical inversion instability By way of example, if n = 1/2 and an error in the (well-posed problem). Accordingly, we briefly present relaxation modulus of maximally 5% is acceptable, this an experimental arrangement that alleviates this inher- condition can be met by recording data only for times ent difficulty [3.28]. At the same time, this particular larger than t/t0 = 5 − ς0/t0.Sinceς0/t0 is always posi- scheme also allows the simultaneous determination tive the relaxation modulus is within about 5% of the of both the relaxation and creep properties, thus cir- ramp-induced measurement as long as one discounts cumventing the calculation of one from the other. In atA Part data taken before 5t0. To be on the safe side, one typ- addition, the resulting data provides the possibility of ically dismisses data for an initial time interval equal to a check on the linearity of the viscoelastic data through ten times the ramp rise time. a standard evaluation of a convolution integral. 3.2 In case the time penalty for the dismissal of that Relaxation and/or creep functions can be deter- time range is too severe, methods have been devised mined from an experimental arrangement that incor- that allow for incorporation of this earlier ramp data porates a linearly elastic spring of spring constant ks as delineated in [3.32, 33]. On the other hand, the wide as illustrated in Fig. 3.6, readily illustrated in terms of availability of computational power makes an additional a tensile situation. The following is, however, subject data reduction scheme available: Using a Prony se- to the assumption that the elastic deformations of the ries (discrete spectrum) representation, one evaluates test frame and/or the load cell are small compared to the constant-strain-rate response with the aid of (3.30), those of the specimen and the deformation of the added leaving the individual values of the spectral lines as un- spring. If the high stiffness of the material does not knowns. With regard to the relaxation times one has two warrant that assumption it is necessary to determine options: the contribution of the testing machine and incorpo- rate it into the stiffness k . Similar relations apply for (a) one leaves them also as unknowns, or s a shear stress/deformation arrangement. In the case of (b) one fixes them such that they are one or two per decade apart over the whole range of the meas- urements. The second option (b) is the easier/faster one and pro- vides essentially the same precision of representation as option (a). After this choice has been made, one fits
the analytical expression with the aid of Matlab to the lb measurement results. Matlab will handle either cases (a) or (b). There may be issues involving possible dynamic overshoots in the rate-transition region, because a test machine is not able to (sufficiently faithfully) duplicate the rapid change in rate transition from constant to zero 0 Δlb rate, unless the initial rate is very low. This discrepancy ls is, however, considerably smaller that that associated
with replacing a ramp loading for a step history. ls
Mixed Uniaxial Deformation/Stress Histories Δl Material parameters from measured relaxation or creep Δl 0 data are typically extracted via Volterra integral equa- s tions of the first kind, i. e., of the type of (3.20) or (3.21). A problem arises because these equations Fig. 3.6 Arrangement for multiple material properties de- are ill-posed in the sense that the determination of termination via a single test Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 59 bulk/volume response the spring could be replaced by and the creep compliance can be determined and the a compressible liquid, though this possibility has not determination of the Prony series parameters proceeds been tested in the laboratory, to our knowledge. For without difficulty [3.21–26] a suddenly applied gross extension (compression) of the The additional inherent characteristic of this (hy- spring by an amount Δl = const, both the bar and the brid) experimental–computational approach is that it spring will change lengths according to may be used for determining the limit of linearly vis- coelastic behavior of the material. By determining the Δl t + Δl t = Δl , (3.32) b( ) s( ) two material functions of creep and relaxation simul- where the notation in Fig. 3.6 is employed (subscript taneously one can examine whether the determined ‘b’ refers to the bar and “s” to the spring). The corre- functions satisfy the essential linearity constraint, see spondingly changing stress (force) in the bar is given (3.62)–(3.64) by t
D(t − ξ)E(ξ)dξ = t . (3.37) A Part Fb(t) = Fs(t) = kb(t)Δls(t) = ks[Δl − Δlb(t)] , 0 (3.33) which is also determined by Time-Harmonic Deformation 3.2 t A frequently employed characterization of viscoelastic = Ab − ξ d [Δ ξ ] ξ materials is achieved through sinusoidal strain histories Fb(t) E(t ) lb( ) d ω lb dξ of frequency . Historically, this type of material char- 0 acterization refers to dynamic properties, because they are measured with moving parts as opposed to methods + Ab Δ 0 , lb E(t) (3.34) leading to quasi-static relaxation or creep. However, in lb the context of mechanics dynamic is reserved for situ- which, together with (3.32), renders upon simple ations involving inertia (wave) effects. For this reason, manipulation we replace in the sequel the traditional dynamic (proper- Δlb(t) Ab ties) with harmonic, signifying sinusoidal. Whether one + εb(0)E(t) Δl ksΔl asks for the response from a strain history that varies with sin(ωt)orcos(ωt) may be accomplished by dealing t d with the (mathematically) complex counterpart + E(t − ξ) [εb(ξ)]dξ = 1 . (3.35) dξ ε(t) = ε0 exp(iωt) · h(t) (3.38) 0 so that after the final statement has been obtained one This is a Volterra integral equation of the second kind, would be interested, correspondingly in either the real as can be readily shown by the transformation of vari- or the imaginary part of the result. Here h(t)isagain ables ξ = t − u; it is well behaved for determining the the Heaviside step function, according to which the real relaxation function E(t). part of the strain history represents a step at zero time By measuring Δl (t) along with the other param- b with amplitude ε . The evaluation of the appropriate re- eters in this equation, one determines the relaxation 0 sponse may be accomplished with the general modulus modulus E(t). representation so that substitution of (3.22)and(3.38) Similarly, one can cast this force equilibrium equa- into (3.12)or(3.13) renders, after an interchange in the tion in terms of the creep compliance of the material and order of integration, the force in the spring as ∞ t dς τ t = ε μ∞ + ς −t/ς kslb d ( ) 2 0 H( )exp( ) ς Fb(t) + D(t − ξ) [Fb(ξ)]dξ Ab dξ 0 0 ∞ t t − ξ + 2ε0iω H(ς) exp − + Fb(0)D(t) = ksΔl . (3.36) ς 0 0 It is clear then that, if both deformations and the stress dς ×exp(iωξ)dξ in the bar are measured, both the relaxation modulus ς 60 Part A Solid Mechanics Topics
t The last term is the transient. An exemplary presenta- μ /μ∞ = ωζ = ζ = + 2ε0iωμ∞ exp(iωξ)dξ, (3.39) tion with s 5, 0 1, and 0 20 is shown in Fig. 3.8. For longer relaxation times the decay lasts 0 longer; for shorter ones the converse is true. One readily which ultimately leads to establishes that in this example the decay is (exponen- tially) complete after four to five times the relaxation τ = ε [μ − μ ] (t) 2 0 (t) ∞ time. The implication for real materials with very long ∞ iωH(ς) relaxation times deserves extended attention. The ex- − 2ε exp(−t/ς)dς pression for the standard linear solid can be generalized 0 1 + iως 0 by replacing (3.41) with the corresponding Prony series ∞ representation. ω ς + ε μ + i H( ) ς . τ(t) 1 ωζ μ 2 (t) ∞ d (3.40) = n n ωt + ωζ ωt 1 + iως 2 2 (cos n sin ) atA Part μ∞ε μ∞ + ω ζ 0 2 0 n 1 n 1 ωζnμn − /ζ The first two terms are transient in nature and (even- − e t n0 . (3.43) μ + ω2ζ 2 tually) die out, while the third term represents the ∞ 1 n 3.2 n steady-state response. Upon noting that the fractions in the last term sum do For the interpretation of measurements it is impor- not exceed μn/2 one can bound the second sum by tant to appreciate the influence of the transient terms on 1 ωζnμn − /ζ the measurements. Even though a standard linear solid, e t n0 μ∞ + ω2ζ 2 represented by the spring–dashpot analog in Fig. 3.7 n 1 n does not reflect the full spectral range of engineering 1 − /ζ 1 μ(t) ≤ μ t n0 = − . materials, it provides a simple demonstration for the de- n e 1 (3.44) 2μ∞ 2 μ∞ cay of the transient terms. Its relaxation modulus (in n shear, for example) is given by This expression tends to zero only when t →∞, a time frame that is, from an experimental point of view, too μ(t) = μ∞ + μs exp(−t/ζ0) , (3.41) long in most instances. For relatively short times that fall into the transition range, the ratio of moduli is not where ζ denotes the relaxation time and μ∞ and μ 0 s small, as it can be on the order of 10 or 100, or even are modulus parameters. Using the imaginary part of larger. There are, however, situations for which this er- (3.40) corresponding to the start-up deformation history ror can be managed, and these correspond to those cases ε(t) = ε sin(ωt)h(t) one finds for the corresponding 0 when the relaxation modulus changes very slowly dur- stress history ing the time while sinusoidal measurements are being τ(t) μs ωζ0 = R = (cos ωt + ωζ0 sin ωt) μ∞ε μ∞ + ω2ζ 2 2 0 1 0 R μ ωζ 6 s 0 −t/ζn − e . (3.42) 5 μ∞ 1 + ω2ζ 2 0 4 3 F 2 1 0 –1 –2 μ η –3 –4 –5 –6 μ0 0 20 40 60 80 100 120 140 160 Normalized time
F Fig. 3.7 Standard Fig. 3.8 Transient start-up behavior of a standard linear linear solid solid under ε(t) = h(t)sin(ωt) Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 61 made. This situation arises when the material is near the complex modulus μ∗ = μ (ω) + iμ (ω) with its real its glassy state or when it approaches rubbery behavior. and imaginary parts defined by ∞ As long as the modulus ratio can be considered nearly 2 (ως) constant in the test period, the error simply offsets the μ (ω) = μ∞ + H(ς)dς (3.47) test results by additive constant values that may be sub- 1 + (ως)2 tracted from the data. Clearly, that proposition does not 0 hold when the material interrogation occurs around the (the storage modulus) and ∞ middle of the transition range. ως There are many measurements being made with μ (ω) = H(ς)dς (3.48) 1 + (ως)2 commercially available test equipment, when frequency 0 scans or relatively short time blocks of different fre- (the loss modulus), respectively. quencies are applied to a test specimen at a set Polar representation allows the shorthand notation temperature, or while the specimen temperature is be- ∗ μ = μ(ω)exp[iΔ(ω)] , (3.49) A Part ing changed continuously. In these situations the data where reduction customarily does not recognize the transient nature of the measurements and caution is required so as μ (ω)
Δ ω = 3.2 tan ( ) μ ω and not to interpret the results without further examination. ( ) Because viscoelastic materials dissipate energy, μ ω ≡|μ∗ ω |= [μ ω ]2 +[μ ω ]2 , prolonged sinusoidal excitation generates rises in ( ) ( ) ( ) ( ) (3.50) temperature. In view of the sensitivity of these mater- so that, also ials to temperature changes as discussed in Sects. 3.2.7 μ (ω) = μ(ω)cosΔ(ω)and and 3.2.8, care is in order not to allow such thermal μ (ω) = μ(ω) sin Δ(ω) . (3.51) build-up to occur unintentionally or not to take such changes into account at the time of test data evaluations. The complex stress response (3.45) can then be written, Consider now only the steady-state portion of (3.40) using (3.50), as so that ∗ τ(t) = 2ε(t)μ (ω) = 2ε0 exp(iωt)μ(ω)exp[iΔ(ω)] , ∞ τ(t) iωH(ς) (3.52) = μ∞ + dς. (3.45) 2ε(t) 1 + iως which may be separated into its real or imaginary part 0 according to
Both the strain ε(t) and the right-hand side are complex τ(t) = 2ε0μ(ω)cos[ωt + Δ(ω)] and numbers. One calls τ(t) = 2ε0μ(ω) sin[ωt + Δ(ω)] . (3.53) ∞ Thus the effect of the viscoelastic material properties ∗ iωH(ς) μ ω ≡ μ∞ + ς is to make the strain lag behind the stress (the strain is ( ) + ως d (3.46) 1 i retarded) as illustrated in Fig. 3.9. It is easy to verify 0 that the high- and low-frequency limits of the steady- state response are given by μ∗(ω →∞) = μ(t → 0) = ∗ τ t μ0, the glassy response, and μ (ω → 0) = μ(t →∞) = ( ) ωt Δ ω Stress: sin[ + ( )] μ∞ 2ε0 μ(ω) , as the long-term or rubbery response (real). ε t ( ) ωt An Example for a Standard Linear Solid. For the stan- Strain: ε = sin 1 0 dard linear solid (Fig. 3.7) the steady-state portion of the response (3.52) simplifies to ω2ς2 0 t μ (ω) = μ∞ + μ , s + ω2ς2 1 0 Δω ως0 μ (ω) = μ , (3.54) s + ω2ς2 1 0 ως0 Fig. 3.9 Illustration of the frequency-dependent phase Δ ω = . (3.55) tan ( ) 2 shift between the applied strain and the resulting stress μ∞/μs + (1 + μ∞/μs)(ως0) 62 Part A Solid Mechanics Topics
which, upon using the transformation t − ξ = u, yields Log of functions 3 ∞ τ(t) − ω ∗ = iω μ(u)e i u du = μ (ω) . (3.59) 2ε(t) 2 −∞
1 If one recalls that the integral represents the Fourier transform F {μ(t), t → ω} of the modulus in the inte- grand one may write 0 ∗ μ (ω) = iωF {μ(t), t → ω} (3.60) –1 along with the inverse, ∞ atA Part –2 ∗ –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 1 μ (ω) − ω μ(t) = e i t dω. (3.61) Log frequency 2π iω −∞ Fig. 3.10 Steady-state response of a standard linear solid to 3.2 sinusoidal excitation, (μ0 = 1, μ = 100, μ/η = ς0 = 0.1). Thus the relaxation modulus can be computed from Symbols: μ (ω); short dash: μ (ω); long dash:tanΔ(ω) the complex modulus by the last integral. Note also that because of (3.60) μ and μ are derivable from a single While this material model is usually not suitable for rep- function, μ(t), so that they are not independent. Con- resenting real solids (its time frame is far too short), versely, if one measures μ and μ in a laboratory they this simple analog model represents all the proper limit should obey a certain interrelation; a deviation in that responses possessed by a real material, in that it has respect may be construed either as unsatisfactory exper- short-term (μ0 + μs, glassy), long-term (μ∞, rubbery) imental work or as evidence of nonlinearly viscoelastic as well as transient response behavior as illustrated behavior. in Fig. 3.10. Note that, with only one relaxation time present, the transition time scale is on the order of at Relationships Among Properties most two decades. The more general representation of In Sect. 3.2.2 exemplary functional representation of the viscoelastic functions under sinusoidal excitation some properties has been described that are generic can also be interpreted as a Fourier transform of the for the description of any viscoelastic property. On the relaxation or creep response. other hand, the situation often arises that a particular function is determined experimentally relatively read- Complex Properties as Fourier Transforms. It is often ily, but really its complementary function is needed. The desirable to derive the harmonic properties from mono- particularly simple situation most often encountered is tonic response behaviors (relaxation or creep). To effect that the modulus is known, but the compliance is needed this consider the strain excitation of (3.38), (or vice versa). This case will be dealt with first. ε = ε ω , Consider the case when the relaxation modulus (t) 0 exp(i t)h(t) (3.56) (in shear), μ(t) is known, and the (shear) creep com- and substitute this into the convolution relation for the pliance J(t) is desired. Clearly, the modulus and the stress (3.11), compliance cannot be independent material functions. In the linearly elastic case these relations lead to re- t dε(ξ) ciprocal relations between modulus and compliance. τ(t) = 2ε(0)μ(t) + 2 μ(t − ξ) dξ, (3.57) dξ One refers to such relationships as inverse relations or 0− functions. Analogous treatments hold for all other vis- and restrict consideration to the steady-state response. coelastic functions. Recall (3.10)or(3.11), which give In this case, the lower limit is at t →−∞so that the the shear stress in terms of an arbitrary strain history. integral may be written as In the linearly elastic case these inverse relations lead to reciprocal relations between modulus and compli- t ance. Recall also that the creep compliance is the strain iωξ τ(t) = lim 2ε0iω μ(t − ξ)e dξ, (3.58) history resulting from a step stress being imposed in t→∞ −t a shear test. As a corollary, if the prescribed strain Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 63 history is the creep compliance, then a constant (step) Thus in the frequency domain of the harmonic mater- stress history must evolve. Accordingly, substitution of ial description the interconnection between properties the compliance J(t)into(3.10) must render the step is purely algebraic. Corresponding relations for the bulk stress of unit amplitude so that behavior follow readily from here. t dJ(ξ) 3.2.7 Dissipation h(t) = J(0)μ(t) + μ(t − ξ) dξ. (3.62) dξ Under Cyclical Deformation 0+ Note that, as t → 0+, J(0+)/μ(0+) = 1 so that at time In view of the immediately following discussion of the 0+ an elastic result prevails. Upon integrating both sides influence of temperature on the time dependence of vis- of (3.62) with respect to time – or alternatively, us- coelastic materials we point out that general experience ing the Laplace transform – one readily arrives at the tells us that cyclical deformations engender heat dis- sipation with an attendant rise in temperature [3.34,
equivalent result; the uniaxial counterpart has already A Part been cited effectively in (3.37). 35]. How the heat generated in a viscoelastic solid as a function of the stress or strain amplitude is described t in [3.13]. Here it suffices to point out that the heat μ(t − ξ)J(ξ)dξ = t . (3.63) generation is proportional to the magnitude of the imag- 3.2 0 inary part of the harmonic modulus or compliance. For this reason these (magnitudes of imaginary parts of the) Note that this relation is completely symmetric in the properties are often referred to as the loss modulus or sense that, also, the loss compliance. We simply quote here a typical t result for the energy w dissipated per cycle and unit vol- J(t − ξ)μ(ξ)dξ = t . (3.64) ume, and refer the reader to [3.13] for a quick, but more 0 detailed exposition: ∞ Similar relations hold for the uniaxial modulus E(t) 2mπ w/cycle = π m ε2 μ . (3.68) and its creep compliance D(t), and for the bulk modulus m T m=1 K(t) and bulk compliance M(t).
Interrelation for Complex Representation. Because 3.2.8 Temperature Effects the so-called harmonic or complex material characteri- zation is the result of prescribing a specific time history Temperature is one of the most important environmental with the frequency as a single time-like (but constant) variables to affect polymers in engineering use, pri- parameter, the interrelation between the complex mod- marily because normal use conditions are relatively ulus and the corresponding compliance is simple. It close to the material characteristic called the glass- follows from equations (3.45)and(3.46)that transition temperature – or glass temperature for short. In parochial terms the glass temperature signifies the iωt 2ε(t) 2ε0 e 1 ∗ temperature at which the material changes from a stiff = = = J ω , (3.65) i(ωt+Δ(ω)) ∗ ( ) τ(t) τ0 e μ (ω) or hard material to a soft or compliant one. The ma- where the function J∗(ω) is the complex shear com- jor effect of the temperature, however perceived by the pliance, with the component J (ω) and imaginary user, is through its influence on the creep or relaxation component −J (ω) related to the complex modulus by time scale of the material. Solids other than polymers also possess character- ∗ γ ω J (ω) = J (ω) − iJ (ω) = J(ω)ei ( ) istic temperatures, such as the melting temperature in 1 e−iΔ(ω) metals, while the melting temperature in the polymer = = (3.66) context signifies specifically the melting of crystallites μ∗(ω) μ(ω) in (semi-)crystalline variants. Also, typical amorphous so that, clearly, solids such as silicate glasses and amorphous metals 1 exhibit distinct glass-transition temperatures; indeed, J(ω) = and γ(ω) =−Δ(ω) , μ(ω) much of our understanding of glass-transition phenom- ena in polymers originated in understanding related ω = [ ω ]2 +[ ω ]2 . with J( ) J ( ) J ( ) (3.67) phenomena in the context of silicate glasses. 64 Part A Solid Mechanics Topics
The Entropic Contribution μ = 1 μ / = obeys ∞ 3 E∞. Thus ∞ T Nk is a material con- Among the long-chain polymers, elastomers possess stant, from which it follows that comparative moduli a molecular structure that comes closest to our ideal- obtained at temperatures T and T0 are related by ized understanding of molecular interaction. Elastomer μ = T μ is an alternative name for rubber, a cross-linked poly- ∞|T ∞|T0 or equivalently mer that possesses a glass transition temperature which T0 is distinctly below normal environmental conditions. = T . E∞|T E∞|T0 (3.70) Molecule segments are freely mobile relative to each T0 other except for being pinned at the cross-link sites. The If one takes into account that temperature changes af- classical constitutive behavior under moderate defor- fect also the dimensions of a test specimen by changing mations (up to about 100% strain in uniaxial tension) both its cross-sectional area and length, this is taken into has been formulated by Treloar [3.36]. Because this account by modifying (3.71) to include the density ratio constitutive formulation involves the entropy of a de- according to atA Part formed rubber network, this temperature effect of the ρT properties is usually called the entropic temperature ef- μ∞|T = μ∞|T or equivalently ρ T 0 fect. In the present context it suffices to quote his results 0 0 3.2 ρT in the form of the constitutive law for an incompress- E∞| = E∞| , (3.70a) T ρ T0 ible solid. Of common interest is the dependence of the 0T0 stress on the material property appropriate for uniaxial where ρ0 is the density at the reference temperature and tension (in the 1-direction) ρ is that for the test conditions. 1 1 To generate a master curve as discussed below it is σ = NkT λ2 − λ λ λ = , 1 λ subject to 1 2 3 1 therefore necessary to first multiply modulus data by the 3 1 ratio of the absolute temperature T (or ρT, if the densi- (3.69) ties at the two temperatures are sufficiently different) at where λ1, λ2,andλ3 denote the (principal) stretch ra- which the data was acquired, and the reference tempera- tios of the deformation illustrated in Fig. 3.11 (though ture T0 (or ρ0T0). For compliance data one multiplies by not shown for the condition λ1λ2λ3 = 1), the multiplica- the inverse density/temperature ratio. tive factor consists of the number of chain segments between cross-links N, k is Boltzmann’s constant, and Time–Temperature Trade-Off Phenomenon T is absolute temperature. Since for infinitesimal defor- A generally much more significant influence of mations λ1 = 1+ε11, one finds that NkT must equal the temperature on the viscoelastic behavior is experienced elastic Young’s modulus E∞. Thus the (small-strain) in connection with the time scales under relaxation or Young’s modulus is directly proportional to the ab- creep. To set the proper stage we define first the notion solute temperature, and this holds also for the shear of the glass-transition temperature Tg. To this end con- modulus because, under the restriction/assumption of sider a measurement of the specific volume as a function incompressibility the shear modulus μ∞ of the rubber
Volume
λ1
λ2 B λ3 A
Equilibrium line T Temperature Fig. 3.11 Deformation of a cube into a parallelepiped. The g unit cube sides have been stretched (contracted) orthogo- Fig. 3.12 Volume–temperature relation for amorphous nally in length to the stretch ratios λ1, λ2,andλ3 solids (polymers) Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 65 of temperature. Typically, such measurements are made sensitive properties, at least for polymers. For ease with a slowly decreasing temperature, curve A in of presentation we ignore first the entropic tempera- Fig. 3.12, because the rate of cooling has an influence on ture effect discussed. The technological evolution of the outcome. Figure 3.12 shows a typical result, which metallic glasses is relatively recent, so that a limited illustrates that at sufficiently low and high temperatures amount of data exist in this regard. However, new data the volume dependence is linear, with a transition con- on the applicability of the time–temperature trade-off necting the two segments. in these materials have been supplied in [3.12]. More- The glass-transition temperature is defined as the over, we limit ourselves to considerations above the intersection of two linear extensions of the two seg- glass-transition temperature, with discussion of behav- ments roughly in the center of the transition range. As ior around or below that temperature range reserved for also indicated in Fig. 3.12, an increase in the rate of later amplification. cooling causes reduced volume shrinkage as a result Experimental constraints usually do not allow the of the unstable evolution of a molecular microstructure full time range of relaxation to be measured at any atA Part that consolidates with time, curve B in Fig. 3.12. This one temperature. Instead, measurements can typically phenomenon is associated with physical aging [3.37– be made only within the time frame of a certain experi- 44]. In practical terms the lowest – most stable – re- mental window, as indicated in Fig. 3.13. sponse curve is determined basically by the patience of This figure shows several (idealized) segments as 3.2 the investigator, though substantial deviations must be resulting from different temperature environments at measured in terms of logarithmic time units: Relatively a fixed (usually atmospheric) pressure. A single curve little may be gained by reducing the cooling rate from 1 may be constructed from these segments by shifting the to 0.1 ◦C/h. temperature segments along the log-time axis (indicated We turn next to the effect of temperature on the by arrows) with respect to one obtained at a (reference) time scale and present this phenomenon in terms of temperature chosen arbitrarily, to construct the master a relaxation response, say, in shear. The discussion is curve. This master curve is then accepted as the re- generic in the sense that it applies, to the best of the sponse of the material over the extended time range at collective scientific knowledge, to all time- and rate- the chosen reference temperature. Because this time– temperature trade-off has been deduced from physical measurements without the benefit of a time scale of G t log ( ) unlimited extent, the assurance that this shift process Experimental is a physically acceptable or valid scheme can be de- window rived only from the quality with which the shifting or
P = P0
T0 = T3 T1 ⎛ σ 273 ⎛ log0 ⎜ ε T ⎜ (psi) ⎝ 0 ⎝ 5 T3 Temperature (°C) T –30.0 2 –25.0 –22.5 –22.0 log aT T 4 4 –17.5 4 –15.0 –12.5 –7.5 Master curve –5.0 at T3 –2.5 3 5.0 T5
ε0 = 0.05 T1 < T2 < ··· superposition can be accomplished. To examine this region. Two arguments dominate, but they are based quality issue requires that test temperatures are cho- on pragmatic rather than rigorously scientific princi- sen sufficiently closely, and that the measurements vary ples. The first argument states that, even in the transition as widely as feasible over the log-time range to afford region, the polymer chain segments experience locally maximum overlap of the shifted curve segments. elastic behavior in accordance with the theory of rubber The amount of shifting along the log-time axis is elasticity. Accordingly, all curve segments obtained at recorded as a function of the temperature. This function the various temperatures should be multiplied by their is usually called the temperature-dependent shift factor, respective ratios of the reference temperature and the or simply the shift factor for short; it is a material char- test temperature, i. e., T0/T, in the case of modulus acteristic, and is often designated by φT . Figures 3.14 measurements, and with the inverse ratio in the case of and 3.15 illustrate the application of the shift principle compliance measurements, regardless of by how many for a polyurethane elastomer, together with the associ- log-time units the material behavior is removed from ated shift factor φT in Fig. 3.16. the rubbery (long-term) domain. The alternative view atA Part asserts that the entropic correction does not apply in The Role of the Entropic Contribution the glassy state and, accordingly should decrease con- Having demonstrated the shift phenomenon in principle, tinuously from the long-term, rubbery domain as the 3.2 it remains to address the effect of the entropic contribu- glassy or short-term behavior is approached. The rule tion to the time-dependent master response. Recall that by which this change occurs is not established scientif- the entropic considerations were derived in the context ically either, but is typically taken to be linear with the of purely rubbery material behavior, and specifically in logarithmic time scale throughout the transition. Ulti- the absence of viscoelastic effects. Thus any modulus mately one needs to decide on the basis of the precision variation with temperature is established, strictly speak- in the data whether one or the other scheme produces ing, only in the long-term time domain when rubbery the better master curve. The crucial argument in that de- behavior dominates, so that (3.70) applies. Various ar- cision is whether the mutual overlap of the segments guments have been put forward [3.45] to apply a similar derived from measurements at different temperatures reduction scheme to data in the viscoelastic transition provides for the most continuous and smoothest master curve. log (shift factor) The Shift Factor 8 While several researchers have contributed significantly to clarifying the concept and the importance of the 6 log10 (273/T) E (t) (psi) 4 5 Reference temperature 2 4.5 T0 = 0°C 5% strain 4 0 3.5 –2 3 –4 –40 –30 –20 –10 0 10 20 2.5 Temperature Fig. 3.15 Time–temperature shift factor for reducing the 2 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 polyurethane data in Fig. 3.14 to that in Fig. 3.16. t φ ◦ log10 / T (min) Tg =−18 CThesolid line represents the WLF-equation Fig. 3.16 Temperature-reduced uniaxial relaxation modu- − . − ◦ φ = 8 86(T 32 C) − . lus for a polyurethane formulation derived from data in log10 T ◦ 4 06 101.6 + (T − 32 C) Fig. 3.14 and with the shift factors in Fig. 3.15 Mechanics of Polymers: Viscoelasticity 3.2 Linear Viscoelasticity 67 time–temperature superposition principle [3.46–49], it changes with time, namely was the group of Williams, Landel,andFerry [3.50] = dt that has been credited with formulating the time– dt or alternatively that φt(T(t)) temperature relationship through the now ubiquitously t quoted WLF equation. They demonstrated the near dt t = . (3.73) universality of this connection for many diverse poly- φT(T(t)) mers, and provided a physical model for the process 0 in terms of a free-volume interpretation. Plazek [3.51– While such an integration can always be effected 53] has supplied an exemplary demonstration of the numerically, in principle, it is important to note that nearly perfect obeyance of the shift phenomenon the logarithmic time scale calls for careful evaluations for polystyrene and poly(vinyl acetate). Ignoring, for of the integrals with variable time steps, yet with- brevity of presentation, the details of the polymer me- out incurring excessive computation time or inaccurate chanical argumentation, WLF derived on the basis of evaluations resulting from too crude an incrementation atA Part free-volume concepts that above the glass-transition of time [3.55]. temperature the shift factor is given by the relation Time–Temperature Shifting Near and Below the Glass c1(T − Tref) 3.2 a = , (3.71) Transition. The time–temperature shifting above the log( T) − − c2 (T Tref) glass-transition temperature has been presented as ba- where Tref denotes an arbitrarily chosen reference sically an empirical rather than a uniquely explained temperature typically about 50 ◦C above the glass- process, though many researchers firmly trust its va- transition temperature of the polymer under consid- lidity because of extensively consistent demonstration eration. The constants c1 and c2 vary from polymer (see, e.g., [3.45,50]). The applicability of the shift prin- to polymer, but for many take on values around ciple to temperatures near and below the glass transition ∼ ∼ c1 = 8.86 and c2 = 101.6. In terms of the relaxation data has been questioned for many years, but is gradu- in Fig. 3.14, the shift procedure renders the composite ally gaining acceptance with certain provisos. First, no or master curve as shown in Fig. 3.16. functional analytic form has been proposed – uniquely supportive or conflicting – that yields credence to the Time–Temperature Trade-Off effect in terms of some molecular model. Moreover, under Transient Temperature Conditions because phenomena at and below the glass transition While the time–temperature shift principle is observed do not occur with molecular conformations in equi- in the laboratory under different temperatures, which are however constant during the measurements, there E t are many situations in the engineering environment log [ ( )] (Pa) 9.6 in which temperatures vary more or less continuously while creep or relaxation processes occur. To assess how such thermal changes affect the viscoelastic response, 22°C 9.4 35°C Morland and Lee proposed [3.54], following the ideas 50°C promulgated in the practice developed for the silicate 65°C glasses, that the time–temperature shift relation applies 9.2 80°C instantaneously. Let T0 denote the reference tempera- 90°C ture at which the master curve has been established and let T be the temperature at which the material behavior 9 is desired. Then the developments in Sect. 3.2.8 above 100°C state that the time (scale) at a temperature T, and desig- 105°C nated by t , is related to the time (scale) t at the reference 8.8 temperature by 012345 log (t) (s) t t = . (3.72) Fig. 3.17 Relaxation behavior in shear for PMMA at var- φ (T) T ious temperatures in the transition range and below the Instantaneous obeyance to this rule requires then glass transition; the glass-transition temperature is 105 ◦C. that (3.72) apply differentially as the temperature (Material supplier: ACE. After [3.15]) 68 Part A Solid Mechanics Topics rates leading to a more unique adherence to a shift log [ µ(t)] (Pa) 9.2 concept. Figure 3.17 shows relaxation data of poly- methyl methacrylate (PMMA) at various (constant) = ◦ 8.8 temperatures below the glass transition (Tg 105 C for PMMA). This data, shifted to produce the master 8.4 curve in Fig. 3.18, generates a shift factor, that while not fitting the WLF equation (3.71), represents never- 8 theless a reasonably coherent relation [3.15], as shown in Fig. 3.19. Many independent, but not thoroughly doc- 7.6 umented, counterparts have been produced over the last decade, which present equally supportive information 7.2 of a consistent time–temperature superposition applica- tion through and below the glass transition. atA Part 6.8 –10 –5 0 5 10 log (t) (s) 3.2.9 The Effect of Pressure on Viscoelastic Behavior of Rubbery Solids 3.2 Fig. 3.18 Relaxation in PMMA, reduced (shifted) com- mensurate with the shift function in Fig. 3.19. The entropic It is important to recognize that pressure can have correction has not been applied to the vertical axis (Mater- a large effect on the viscoelastic response through ial supplier: ACE. After [3.15]) its influence on the free volume. This fact is im- portant when high speed impact is involved, such librium, the ideas underlying the shift phenomenon as when measurements are made with split Hopkin- above the glass transition are questioned more read- son bars, or when materials are otherwise subjected ily in the context of these lower temperatures. For to high pressures (civil engineering: building sup- example, the role of nonequilibrium changes in free- port pads for protection against earthquake damage). volume interferes with simple concepts and complicates Similar to the time–temperature trade-off, pressure the rules by which such examinations and data in- produces a pressure-sensitive shift phenomenon.Fig- terpretations are carried out. For example, Losi and Knauss [3.56] have argued on the basis of free-volume G t considerations that any shift operation below the glass log ( ) transition should depend on the temperature rate with Experimental which the state of the polymer is approached, slower window T = T0 P P φ t 0 = 3 log [ T ( )] (s) P5 12 E t Shift factor for ( ) P3 10 P4 8 log aP 2 6 P2 4 Master curve P 2 at 3 P1 0 P P P –2 1 < 2 < ··· < 5 0 25 50 75 100 125 Temperature (°C) log t Fig. 3.19 Shift factor derived from Fig. 3.17 to generate the Fig. 3.20 Effect of pressure on the viscoelastic response of master curve in Fig. 3.18 (after [3.15]) a polymer (after [3.57]) Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 69 ure 3.20 illustrates this material behavior parallel to The 00 superscript indicates that the parameter is re- the time–temperature shift phenomenon. Without delv- ferred to the reference temperature T0 (first place) and ing into the molecular reasoning for modeling this to the reference pressure P0 (second place). A sin- phenomenon [3.58] we quote here the results by gle 0 superscript refers to the reference temperature Fillers, Moonan,andTschoegl [3.59–62], which ex- only. The asterisk superscript refers to zero pressure. tend the temperature shift factor into a consolidated Equation (3.76) is the Fillers–Moonan–Tschoegl (FMT) temperature-and-pressure shift factor of the form (note equation. Setting θ(P) = 0 (i. e., performing the exper- the similarity to the WLF equation (3.71)) iments at the reference pressure), the FMT equation B reduces to the WLF equation (3.71). log α =− T,P 2.303 f 0 3.2.10 The Effect of Moisture and Solvents T − T0 − θ(P) × , (3.74) on Viscoelastic Behavior f0/αf(P) + T − T0 − θ(P) where It has been observed that the presence of mois- A Part θ = /β ture in some polymers has an effect similar to that (P) fT0 (P) f(P)and P P of an elevated temperature in that increased mois- 3.3 ture content shortens the relaxation or retardation = κ − κφ (3.75) fT0 (P) eT0 dP T0 dP times [3.63, 64]. This process may or may not be re- P0 P0 versible. Plazek [3.52] has pointed out, for example, with κe denoting the compressibility of the entire vol- that in polyvinyl acetate moisture must be removed ume, and κφ is the compressibility of the occupied carefully prior to forming centimeter-sized test sam- volume. ples. Once such larger test samples have absorbed ∗ (some) moisture, it may be impossible to totally re- If we let Ke (T) be the bulk modulus at zero pressure and κe a proportionality constant, one arrives at move the same. In determining viscoelastic properties 00[ − − θ ] it is thus important that one assess the tendency of c1 T T0 (P) log αT,P =− with the material to absorb moisture or other solvents. c00(P) + T − T − θ(P) 2 0 This property can be disturbing during the acquisi- 1 + c0 P 1 + c0 P tion of mechanical properties if no specific precautions θ = 0 4 − 0 6 (P) c3(P)ln c5(P)ln are taken: For example, measurements performed on 1 + c0 P 1 + c0 P 4 6 days during changing humidity may render data that (3.76) violates the concept of time–temperature trade-off. and the notation A technologically important material with a con- 00 = / . ; 00 = /β ; c1 B 2 303 f0 c2 f0 f(P) siderable tendency to absorb moisture is nylon (6 ∗ and 66), with corresponding implications for the de- c00 = 1/k β (P) ; c00 = k /K ; 3 r f 4 e e formability and/or structural load-carrying ability over 00 = / β ; 00 = / ∗ . c5 1 kφ f(P) c6 kφ Kφ (3.77) time. 3.3 Measurements and Methods A considerable range of commercial equipment has (temperature and moisture control). Differential thermal been developed over the years to characterize vis- analyzers (DTAs) are available commercially to mea- coelastic material behavior. Such instrumentation for sure glass transition and melt temperatures, though they tensile and compression tests are standard screw-type are sometimes combined with force measurement ca- test frames (Instron, Zwick) or servohydraulic machines pability, and these devices often function with short, (MTS, Instron); the latter type are available for torsional stubby bend specimens and, as typically used, provide (shear) and combined tensile/torsional characteriza- more qualitative rather than precise property meas- tion, also. Because of the sensitivity to environmental urements. Similarly, dynamical mechanical analyzers temperatures, these machines are usually equipped – (DMAs) are more geared to making mechanical prop- or should be – with environmental control chambers erties measurements and can employ either steady-state 70 Part A Solid Mechanics Topics loading or oscillatory excitation for frequency sensitive quick and rough estimation of physical properties are properties, though here the same caveat is in order as for desired, careful measurements demand absolute envi- work with DTAs. ronmental control that can be afforded only through For frequency imposed shear deformations (rota- suitable environmental chambers, most often matched tion) commercial test equipment is available and acces- to existing or purchased test frame systems. How sible over the web (e.g., Google→Rheometrics). Except closely environmental control must be exercised de- for nanoindentation equipment, to be discussed later, pends on the study at hand: Thermal control ranges other instrumentation is constructed for specific tasks. that may be necessary can be estimated with the aid of For example, Plazek has provided an exemplary con- the shift factor delineated in Sect. 3.2.8. Moisture con- struction of long-term measurement equipment [3.65] trol may be exercised by way of saline solutions if the that provides precise force/moment definition and test chamber is relatively small, or by injecting suit- recording equipment that is unusually stable over long ably proportioned streams of dry and water saturated periods approaching six decades of time. A thought- air into the (larger) test chamber [3.63]. The degree to atA Part ful design of a torsion pendulum for shorter time which moisture influences the time-dependent behav- frames has been supplied by the same researcher [3.66]. ior may be estimated from volume considerations if A rheometer utilizing an eddy-current torque transducer the swelling from moisture of the material is known 3.3 and an air-bearing suspension has been developed by (it may have to be measured separately). One esti- Berry et al. [3.67] for measurements of creep func- mates that the moisture-induced volume change equals tions in shear in both the time and frequency domains. a thermally induced volume change, from which con- For investigating the interaction of shear and volume sideration one deduces the influence of moisture via response, Duran and McKenna [3.68] developed a tor- Sect. 3.2.8 in the form of an equivalent temperature siometer with ultrafine temperature control to access sensitivity. From this information one may estimate time-dependent changes in volume resulting from the whether the potential moisture influence is disturbingly torsion of a cylinder, with the volume change being large. monitored via a mercury column. The Use of Wire and Foil Strain Gages 3.3.1 Laboratory Concerns on Polymers It is clear on general principles that the use of wire/foil While test procedures for typical material characteriza- gages is ill advised for soft materials because one typ- tion have been in place in the laboratory for many years, ically needs to ignore the effect of reinforcing the test there are special considerations that apply to the de- material by the stiff, if thin, gage(s). While it has been termination of viscoelastic properties. Because usually extensively attempted to develop strain-measuring de- long-term measurements are in order, careful attention vices with foil gages for application to relatively soft needs to be paid to temporal consistency of the equip- solids (solid propellant rocket fuels) with the aid of ment. The most important and often recurring issues are computational and experimental analysis tools, these discussed here briefly. costly programs have not yielded useful results. In contrast, wire/foil gages are, however, used on rigid Equipment Stability polymers. Because often extensive time intervals are needed to As has been demonstrated amply in the above de- record data, the associated electrical equipment needs to velopments, polymer mechanical responses can be very be commensurately stable for long periods of time. Of- sensitive to temperatures. This fact is important to ten electronic equipment will record data without drift remember in connection with the use of electric-current- for periods of an hour or two. However, with the desire driven strain gages bonded to polymers. Because these to record data for as long as days, one needs to be as- devices dissipate heat in immediate proximity to the sured that during that time interval the equipment does polymer, this phenomenon needs to be controlled. not drift or does not do so to a significant extent. The While it is true that in the past strain gages have been easiest way of checking stability is by trial. bonded to metallic structures without much concern for the thermal effects on the bonding agent, it must Environmental Control be remembered that these bonding agents had been All polymers are sensitive to thermal variations and developed with this special concern in mind, being many also to moisture changes. While sometimes only also well aware that the metallic component usually Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 71 provided excellent heat conductivity. Even so, it is rise to jaw flow. The traditional way to cope with this a common, if not universal, practice to activate the elec- type of occurrence is to bond metal (aluminum) tabs to tric current in the strain gage(s) only intermittently so the ends of a specimen so as to redistribute the clamping as to reduce the heat generation. If foil gages are de- forces. sired, the sufficiency of this option must certainly be considered. Mechanical Overshoot Phenomenon The overriding consideration in this connection is When relaxation tests are of interest the usual the temperature achieved at the strain gage site rela- test machines provide a ramp history, as discussed tive to the glass-transition temperature of the polymer, in Sect. 3.2.6. If examined in detail these ramp histories on the one hand, and the duration of the measurements typically exhibit an overshoot phenomenon that derives on the other. Thus, a strain gage may well serve on from the dynamics (inertia) of the test machine. Re- a polymer in a wave propagation experiment, but may call that it is usually of interest to gain access to as fail miserably in similar circumstances if the test time large a test duration as possible. Basically three avenues atA Part is measured in weeks. As a quick rule of thumb it is are open to the investigator, depending on the need for our recommendation that temperatures under a gage re- precision and necessary time range: main 50 ◦C below the glass-transition temperature for applications of short duration. Ultimately, one must 1. The time history involving the loading transients are 3.3 consider the relaxation or creep behavior of the polymer ignored by disregarding the initial time history ex- for the range of temperatures anticipated in the exper- tending over ten times the ramp rise time. This is iment, whereby a fractional change in stiffness must a serious experimental restriction, but represents the be estimated for the expected duration of the meas- method most consistently practised in the past. urements. Clearly, one needs to be concerned with both 2. Expand the initial time scale by resorting to the the temperature and the test duration. This estimation method described in [3.32]or[3.33], remaining may actually require that the temperature in the gage conscious, however, that the overshoot between the vicinity be determined experimentally (infrared tooling) linear rise and constant deformation history should and the results coupled with a numerical stress analysis not interfere with the assumptions underlying these assessment of the effect on the gage readout(s). Here approximations. This means that the deformation it should also be remembered that modulus and com- history has to be carefully recorded. pliance data are typically presented on a logarithmic 3. Write out in closed form the response to the full scale, making changes due to temperature appear small, ramp history and fit this analytical expression to the when in fact the true change is considerably larger. measurement data. With a Prony series representa- An error on the order of 10–15% in misinterpreted tion it is advisable to choose a series representation modulus data could translate into a commensurately that contains at least as many terms as are desired large and systematic error in the ultimate, experimental for the test duration. While this may be a tedious results. algebraic process, it may not be easily possible to achieve the same goal without a Prony-series repre- Soft Materials sentation. The overshoot phenomenon would then Most of the more standard engineering materials are rel- require a judicious redacting of the data so as to atively solid or stiff, so that clamping a specimen in the eliminate the inaccuracies derived from it. jaws of a typical test machine poses no particular prob- 4. The final, most precise method to date would entail lem. However, many polymers are either relatively soft a careful measurement of the deformation history, (to the touch) or when stiff nevertheless will creep out including the overshoot modeled by an integrable of the machine gripping device(s) over time. Because function, and apply the same representation method the expected response may be a decrease over time (as indicated under (3) above. in a relaxation test), a slow flow of material out of the gripping jaws may not become apparent unless careful 3.3.2 Volumetric (Bulk) Response tracking of that potential process is achieved. This prob- lem is most prevalent, for example, when dealing with In this section we distinguish between methods near-rubbery behavior under tension, because tension geared to determining small-strain volumetric proper- invokes Poisson contraction, which is maximal under ties and those derived from non-infinitesimal volumetric these conditions (ν = 1/2) and thus most prone to give deformations. 72 Part A Solid Mechanics Topics Very Small Volumetric Strains pick-up measures the pressure response with respect In linearly elastic materials bulk behavior is most typ- to both amplitude and phase shift relative to the input ically determined (a) directly from wave propagation pressure. The compressibility of the liquid having been measurements or (b) indirectly from shear or Young’s determined by calibration, the specimen compressibil- modulus data together with Poisson’s ratio as meas- ity modifies the cavity signal (by a small amount). From ured, e.g., from the antielastic curvature of beams. For the (complex) difference one derives the harmonic bulk viscoelastic solids other methods need to be employed modulus or compliance. Because these are difference with nontrivial equipment requirements. In our experi- measurements, the precision for the bulk behavior re- ence, this method has not been applied to viscoelastic quires the ultimate in precision in instrumentation and materials. The dominant reason is, most likely, that the calibration. A major limitation of this approach is that formation of an optical gap between the specimen and because of potential resonances the range of frequencies the reference mirrors, which would rest on the speci- is also limited to less than four decades of frequency men edges, must cope with the deformations generated (time). For further detail the reader may wish to consult atA Part by the weight of the reference mirror. Absent a lo- the references [3.71, 72]. cal reference mirror an interferometric assessment of the deformation field would require the subtraction of Bulk Measurements Allowing 3.3 the overall specimen deformation from the curvature- also for Non-infinitesimal Volume Strains inclusive deformation pattern, a process that is prone An alternative method, though associated typically with to lead to relatively large errors. For viscoelastic solids larger volume strains, has been offered successfully other methods need to be employed with non-trivial by Ma and RaviÐChandar [3.73, 74], Qvale and Ravi- equipment requirements. Because the effort is consider- Chandar [3.75], and Park et al. [3.76], who followed the able (see the historical development starting with [3.69, same method. This method involves a hollow cylinder 70]) the details of this measurement method are not pre- instrumented with strain gages on its exterior surface. sented here, other than to point out via Fig. 3.21 what is A physically closely fitting solid specimen is formed in, involved in principle. or introduced into, its interior and pressure is applied As illustrated in Fig. 3.21, one process revolves through an axially closely fitting compression piston, around a stiff cavity to receive a specimen, after which as illustrated in Fig. 3.22. Proper choice of the cylin- the cavity is filled with an appropriate liquid with which der material and its wall thickness allows optimization the specimen does not interact (by swelling or oth- of the response measurements to, say, a constant pis- erwise). A piezoelectric driver generates (relatively) ton displacement (bulk relaxation) from which the bulk small sinusoidal pressure variations, which compress relaxation modulus can then be determined. If the cylin- the specimen in a quasistatic manner as long as the cav- der is manufactured from a material that remains elastic ity size is chosen appropriately. A separate piezoelectric during a test (say steel), with a and b denoting its inter- a)Oil outlet Electrode b) Electrical Teflon washer feed-through Outlet needle valve Oil outlet Cavity Piezoelectric disk Specimen Grid for specimen K-Seal support Inlet needle valve Oil inlet Electrode Oil inlet Fig. 3.21 (a) Global and (b) local cavity arrangement for measuring the bulk modulus with harmonic excitation Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 73 2 ezz(t) = [ezz(t) − err (t)] , 3 1 err (t) = eθθ (t) =− [ezz(t) − err (t)] , (3.81) (Steel) 3 loading pins Confining which are connected by the constitutive relations cylinder t ∂δ(ξ) σ (t) = 3σ (t) = 3 K(t − ξ) dξ, (3.82) Strain gage(s) kk m ∂ξ Specimen −∞ t ∂δ(ξ) s t = μ t − ξ ξ, (3.83) ij( ) 2 ( ) ∂ξ d −∞ atA Part from which the bulk modulus K(t) and the shear modu- lus μ(t) can be determined. Typically, a constant piston displacement can be Fig. 3.22 Cylinder/piston arrangement for determining used to lead to relaxation behavior or alternatively, 3.3 bulk response a constant relative velocity of the pistons can be used in the last set of equations. Ravi-Chandar and his co- nal and external radius, respectively, the circumferential workers demonstrated axial strains of as high as ≈ 20%, strain εθ on the exterior surface is related to the internal though the deformation in the linearly viscoelastic do- pressure σrr and the strains on the cylindrical specimen main required only strains on the order of 5% or less. surface εrr = εθθ by These latter values are still much larger than those en- / 2 − countered in the harmonic test method [3.71,72] though (b a) 1 c σrr (t) = σθθ (t) =− E εh , the same results should prevail with this difference in 2 2 magnitudes, as long as one is convinced that the linear 1 c c b ε t = εθθ t = ε t − v + + v , properties extend over this larger strain range. Because rr ( ) ( ) h( ) (1 ) (1 ) 2 2 a the specimen deformation depends on both the bulk and σzz(t) = σa(t) , on the shear characteristics of the material one can eval- εzz(t) = εa(t) , (3.78) uate simultaneously the (relaxation) shear modulus as where the usual nomenclature of radial coordinates ap- well. A constant axial piston velocity may be used as an plies, Ec and vc are the elastic properties of the (steel) alternative loading history. cylinder, and the subscript ‘a’ refers to the axially ori- ented stress and strain as determined, respectively, from log (modulus/GPa) the load cell of the test frame or measured by the relative 0.5 motion of the pressure pistons. Upon expressing the stress and strain fields in the 0 specimen into dilatational and deviatoric (shear) com- ponents by using the mean stress –0.5 1 σm(t) = [σzz(t) + 2σrr (t)] (3.79) 3 –1 and the dilatation Bulk modulus Shear modulus δ(t) = εzz(t) + 2εrr (t) . (3.80) –1.5 Along with the usual definition of deviatoric compo- –2 nents of stress sij(t) and strain eij(t), –5–4–3–2–10123456 log t 2 szz(t) = [σzz(t) − σrr (t)] , 3 Fig. 3.23 Bulk and shear modulus in relaxation both ob- 1 tained in a single measurement series in the apparatus of srr (t) = sθθ (t) =− [σzz(t) − σrr (t)] , 3 Fig. 3.22 74 Part A Solid Mechanics Topics the same time dependence, Poisson’s ratio would be E t ( ) (GPa) a constant. From these figures it is immediately appar- 10 ent over what range the bulk and shear moduli exhibit Highly closely the same time dependence and over what range confined the approximation of a constant bulk modulus would 1 render acceptable or even good results in a viscoelastic Confined analysis. Unconfined Qvale and Ravi-Chandar [3.75] point out that the polymer is well below the glass-transition temperature 0.1 and that the effect of moving to high pressures is to ex- Tref = 80°C tend the relaxation or retardation times to longer values as would be the result of cooling the material to lower 0.01 temperatures. An example of this effect is demonstrated atA Part –4 –2 0 2 4 6 10 10 10 10 10 10 in Fig. 3.24 [3.75], which shows three data sets: one, t/a T identified as unconfined is for zero superposed pressure; Fig. 3.24 Uniaxial relaxation modulus as determined in the other two result from different degrees of pressure 3.3 the configuration of Fig. 3.22, the confinement being con- as controlled by the choice of material for the confining trolled by the stiffness of the exterior cylinder [3.75] cylinder Fig. 3.22. This effect is thought to be linked to the reduction in free volume of the polymer as a result Special care is required in order to minimize or of the compression. eliminate the gap between the specimen and the interior cylinder wall. Because of the typically relatively high 3.3.3 The CEM Measuring System compressibility of polymers, the measurements are sen- sitive to small dimensional changes in the test geometry. A relatively recent measurement system providing for However, sufficient precision can be achieved with ap- the determination of numerous (time-dependent) prop- propriate care, as demonstrated in references [3.73–75]. erties besides bulk response with a high degree of An example of measurement evaluations are shown (Plazek)-precision is the CEM measuring system (taken in Fig. 3.23 for PMMA in the form of both bulk and from the initials of the Center for Experimental Me- shear behavior. If the shear and bulk modulus possessed chanics, University of Ljubljana, Slovenia) [3.57, 77]. Table 3.2 Measuring capabilities of the CEM apparatus Physical Properties Symbols Temperature T(t) Pressure P(t) Measured Angular displacement ϑ0 = ϑ(t = 0) Specimen length L(t), L(T), L(P)orL(t, T, P) Torque M(t), M(T), M(P)orM(t, T, P) Shear relaxation modulus G(t), G(T), G(P)orG(t, T, P) Shear compliance J(t), J(T), J(P)orJ(t, T, P) ν ,ν ,ν ν , , Calculated from Specific volume (t) (T) (P)or(t T P) α ,α α , definitions Linear thermal expansion coefficient (T) (P)or(T P) Volumetric thermal expansion coefficient β(T),β(P),βg,βe,βf or βgef(T, P) Bulk creep compliance B(t), B(T), B(P)orB(t, T, P) Bulk modulus K(T), K(P)orK(T, P) WLF constants c1, c2 α Calculated from WLF material parameters f, f0 models FMT constants c1, c2, c3, c4, c5, c6 α α ∗ FMT material parameters f(P), 0(P), B, Ke , ke, Kφ, kφ Shift factors a(T), a(P) and/or a(T, P) Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 75 Thermal bath Pressure vessel Electromagnet Measuring inserts Silicone oil Carrier amplifier Circulator atA Part Silicone fluid 3.3 Data acquisition Magnet and motor charger Pressurizing system Fig. 3.25 Schematic of the CEM measuring system Although the apparatus is not yet available as a rou- ± 0.1 MPa, and to temperatures ranging from −50 ◦C tine commercial product, we cite here its components to +120 ◦C with a precision of ± 0.01 ◦C. because of the larger-than-normal range of properties that can be determined with it. This list is shown in Ta- The Relaxometer ble 3.2. The system measures five physical quantities: The relaxometer insert, shown in Fig. 3.26a, measures temperature, T(t); pressure, P(t); torsional deformation the shear relaxation modulus by applying a constant (angular displacement) per unit length, ϑ0, applied to torsional strain to a cylindrical specimen, and by mon- the specimen at t = 0; specimen length L(t, T, P); and itoring the induced moment as a function of time. The the decaying torque, M(t, T, P), resulting from the ini- specimen diameter can range from 2 mm to 10 mm, and tial torsional deformation, ϑ0. its length from 52 mm to 58 mm. For details on speci- The system assembly is shown schematically men preparation the reader is referred to [3.77]. in Fig. 3.25. The pressure is generated by the pressur- Two main parts of the insert are the loading de- izing system using silicone oil. The pressure vessel is vice,andtheload cell. The loading device applies contained within a thermal bath, through which another a torsional strain by twisting the specimen a few ◦ silicone oil circulates from the circulator, used for close degrees (typically around 2 in less than 0.01 s, de- control of the temperature. The apparatus utilizes two pending on the initial stiffness of the specimen). To separate measuring inserts, which can be housed in the effect this deformation, the electric motor first preloads pressure vessel: the relaxometer, shown in Fig. 3.26a, a torsion spring. Once twisted, the spring is kept in and the dilatometer, shown in Fig. 3.26b. Signals from its preloaded position by a rack-and-pawl mechanism. these measuring inserts pass through the carrier ampli- The activation of the electromagnet, mounted outside fier prior to being collected in digital format by the data the pressure vessel (Fig. 3.25), releases the pawl so acquisition system. that the spring deforms the specimen to a predeter- The magnet and motor charger supplies power to mined angle. The induced moment is then measured the electromagnet, which initiates the measurement. by the load cell, which is attached to the slider mecha- The same charger also supplies current to the elec- nism to compensate for possible changes in the length tric motor of the relaxometer, shown in Fig. 3.26a, of the specimen resulting from changes in tempera- which preloads the spring that then applies the de- ture, pressure, and the Poynting effect (shortening sired torsional deformation (angular displacement) to of the specimen caused by a torsional deformation). the specimen. Specimens can be simultaneously sub- After the shear relaxation measurement is complete, jected to pressures of up to 600 MPa with a precision of the electric motor brings the specimen to its origi- 76 Part A Solid Mechanics Topics Fig. 3.26 The CEM re- a) b) laxometer and dilatometer Loading inserts device Triggering mechanism LVDT Electric motor 280 mmLVDT rod 260 mm Specimen Slider Specimen mechanism atA Part Load cell 3.3 nal undeformed state, while maintaining the pressure by the weight of the rod. For linearly viscoelastic behav- vessel fully pressurized. The relaxometer can measure ior this limitation can be easily corrected. shear moduli in the range 1–4000 MPa, with a maxi- mal relative error of 1% over the complete measuring 3.3.4 Nano/Microindentation range. for Measurements of Viscoelastic Properties The Dilatometer of Small Amounts of Material The dilatometer insert, shown in Fig. 3.26b, is used to measure bulk properties such as the: bulk creep It is often necessary, as in a developmental research en- compliance, B(t, T, P); equilibrium bulk creep com- vironment, to determine viscoelastic properties when pliance, B(T, P) = B(t →∞, T, P); specific volume only very small or thin specimens are available. The ν(T, P) = ν(t →∞, T, P), and thermal (equilibrium) nanoindentation technique developed over the past two expansion coefficient, β(T, P) = β(t →∞, T, P). The decades [3.78–80] has been demonstrated to be effec- bulk compliance may be inverted to yield the bulk mod- tive in such cases where thin films or microstructural ulus. Measurements are performed by monitoring the domains in homogeneous or inhomogeneous solids are volume change of the specimen which results from the concerned. Methods have been established for the meas- imposed changes in pressure and/or temperature, by urements of properties such as the Young’s modulus measuring the change in specimen length, L(t, T, P), for materials exhibiting time- or rate-insensitive be- with the aid of a built-in linearly variable differential havior. Under the assumption that unloading induces transformer (LVDT). The volume estimate can be con- only elastic recovery, Oliver and Pharr [3.80] pioneered sidered accurate if the change in volume is small (up to a method to measure Young’s modulus of time- or a few percent) and the material is isotropic. Dilatome- strain-rate-independent materials, using an assumed or ter specimens may be up to 16 mm in diameter and known Poisson’s ratio. This method, which is based 40–60 mm in length. The relative measurement error in primarily on Sneddon’s solution [3.81], can measure volume is 0.05%. properties such as Young’s modulus and harness with- Displacement dilatometry has an accuracy advan- out the need to measure directly the projected areas of tage over mercury confinement dilatometry inasmuch permanent indent impressions in an inelastic solids by as it allows an easily automated measurement process employing a modified or equivalent linearly elastic ma- for tracking transient volume changes over extended terial response. While such methods work well for time- periods of time. However, for soft materials an impor- or rate-independent materials (metals, ceramics, etc.), tant limitation arises (usually at temperatures above Tg), applying these methods directly – i. e., without proper when the specimen’s creep under its own weight be- modifications – to viscoelastic materials is not appro- comes significant. Given the arrangement of the LVDT priate. For example, the unloading curve in viscoelastic rod (Fig. 3.26b), there will be an additional creep caused materials sometimes has a negative slope [3.82] under Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 77 tation of a rigid, axisymmetric indenter pressed into a)Z b) a homogeneous, linearly elastic and isotropic half- space. The indentation depth H (Fig. 3.27)ofthe a axisymmetric indenter tip is represented in terms of the R α H indenter geometry by Hc O r H 1 f (x)dx Fig. 3.27 (a) A conical indenter and (b) a spherical indenter H = √ , (3.84) 1 − x2 0 situations where small unloading rates and relatively high loads are used for a material with pronounced vis- where z = f (x) is the shape function for an axisymmet- coelastic effects. ric indenter, with x = r/a being the coordinate shown Accordingly, special procedures have been devel- in Fig. 3.27; the origin of the frame is coincident with atA Part oped in recent years to measure viscoelastic behavior – the indenter tip and a is the radius of the contact circle relaxation modulus and creep compliance – for linearly at the depth Hc. viscoelastic materials. Cheng et al. [3.83] developed According to this analysis the load on an axisym- a method to determine viscoelastic properties using metric indenter is 3.3 a flat-punch indenter. Lu et al. [3.84] proposed meth- 1 ods to measure the creep compliance in the time domain 4Ga x2 f (x)dx P = √ , (3.85) of solid polymers using either the Berkovich or spheri- 1 − ν 1 − x2 cal indenter. Huang et al. [3.85] developed methods to 0 measure the complex modulus in the frequency domain where G and ν are the shear modulus and Poisson’s using a spherical indenter. Some of these methods are ratio, respectively. summarized and discussed below. For a conical indenter one has z = f (x) = ax tan α, Measurements of Viscoelastic Functions so that (3.84) becomes in the Time Domain 1 H = πa tan α, (3.86) Nanoindentation into a bulk material can often be con- 2 sidered as a process of indenting a half-space with a rigid indenter. Typically, indenters are made of di- with the angle α defined as in Fig. 3.27a. The indenta- amond, so that their Young’s modulus is at least tionloadin(3.85)isthengivenby two orders of magnitude greater than that for a typ- πGa2 ical viscoelastic material; the indenter can then be P = tan α, (3.87) considered to be rigid. Figure 3.27 shows a conical 1 − ν indenter and a spherical indenter. In nanoindentation which, upon using (3.86), renders the load–depth rela- testing, a pyramid-shaped indenter is often modeled tion for a conical indenter as as a conical indenter with a cone angle that pro- 4 vides the same area-to-depth relationship as the actual P = GH2 . (3.88) pyramidal indenter. As in the case of the Berkovich π(1 − ν)tanα indenter, it can be modeled as an axisymmetric con- Similarly, for the half-space indentation by a spherical ical indenter with an effective half-cone angle of ◦ indenter, with the geometry shown in Fig. 3.27b, the in- 70.3 . This makes solutions for axisymmetric elastic in- = = − 2 − 2 2 dentation problems available for determining material denter shape function is z f (x) R (R a x ), properties such as Young’s modulus with pyramid- where R is the sphere radius. Substituting into shaped indenters and the (linearly) viscoelastic behavior (3.84)and(3.85), one finds the load–displacement of the material can be determined by way of the relation [3.86] √ load–displacement data obtained from indenting a vis- 8 R / coelastic solid. P = GH3 2 (3.89) 3(1 − ν) Linearly Elastic Indentation Problem. Sneddon [3.81] under condition of a small H/R ratio, typically H/R < derived the load–displacement relation for the inden- 0.2 (see below). 78 Part A Solid Mechanics Topics The Linearly Viscoelastic Indentation Problem. For- P(t) = P0h(t), where P0 is the magnitude of the inden- cing a rigid indenter into a linearly viscoelastic, tation load, and h(t) is the Heaviside unit step function. homogeneous half-space can be treated as a quasistatic Substituting this into (3.90) for a conical indenter one boundary value problem with a moving boundary be- deduces the shear creep function from tween the indenter and the half-space, as the contact 4H2(t) area between the indenter and the half-space changes J(t) = . (3.92) with time. Note that because of the moving bound- π(1 − ν)P0 tan α ary condition the correspondence principle between Similarly, if the indentation load P(t) = P0h(t)is a linearly viscoelastic solution and a linearly elastic applied to a spherical indenter with (3.91) the shear solution is not applicable. To solve this problem, Lee creep function is determined from and Radok [3.87] suggested to find the time-dependent √ stresses and deformations for an axisymmetric inden- 8 RH3/2(t) J t = . (3.93) ter through the use of a hereditary integral operator ( ) − ν atA Part 3(1 )P0 based on the associated solution for a linearly elas- tic material. Applying this Lee–Radok method (e.g., see Riande et al. [3.88]) to (3.88) leads to the time- Indentation under a Constant Load Rate. With 3.3 dependent indentation depth for any load history that Fig. 3.4 in mind we discuss first the case of determining does not produce a decrease in contact area (linearly the creep compliance from the measured load–depth re- lation for a load increasing at a constant rate. Let the viscoelastic material) ˙ ˙ load be P(t) = P0th(t), with P0 being the load rate t (mN/sorμN/s). π(1 − ν)tanα dP(ξ) H2(t) = J(t − ξ) dξ. For a conical indenter, substitution of this P(t)into 4 dξ (3.90) yields 0 (3.90) ˙ t π(1 − ν)P0 tan α H2(t) = J(t − ξ)dξ, (3.94) where J(t) is the shear creep compliance at time t. 4 Radok [3.89], Lee and Radok [3.87], Hunter [3.90], and 0 Yang [3.91] have investigated the indentation into lin- which upon differentiation with respect to time yields early viscoelastic materials with a spherical indenter. 8H(t) dH(t) For a rigid, spherical indenter with radius R, upon using J t = . (3.95) ( ) π − ν α the hereditary integral in (3.90), the relation between (1 )P0 tan dt load and penetration depth is represented by Equation (3.95) determines the shear creep compliance J(t) from the measured indentation depth history H(t). t / 3(1 − ν) dP(ξ) This simplifies (for a constant load rate) to H3 2(t) = √ J(t − ξ) dξ. 8 R dξ 8H(t) dH 0 J t = t . (3.96) ( ) π − ν α ( ) (3.91) (1 )tan dP For a spherical indenter, we have from (3.91) Either (3.90) (conical indenter) or (3.91) (spheri- cal indenter) can be used to determine the shear creep ˙ t / 3(1 − ν)P0 compliance J(t) under a prescribed loading history, as H3 2(t) = √ J(t − θ)dθ. (3.97) illustrated below. We note that Poisson’s ratio is as- 8 R 0 sumed to be constant in the above derivation. In the sequel we describe three monotonic load histories for Differentiation of (3.97) with respect to t yields determining the creep compliance of a linearly vis- √ 4 RH1/2(t) dH coelastic material: (i) a step load, (ii) a constant load = , J(t) ˙ (3.98) rate, and (iii) a ramp loading with an initially constant (1 − ν)P0 dt load rate. or, simplified, √ / Indentation under a Step Load. In the case of a step 4 RH1 2(t) dH J(t) = (t) . (3.99) load applied to the indenter, the load is represented by (1 − ν) dP Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 79 ˙ For loads increasing with constant rates, (3.96)and Since P(t) = P0t again, (3.103) becomes (3.99) are the equations for deriving the creep function − ν N using conical or spherical indenters, respectively. Both 3/2 3(1 ) H (t) = √ J + Ji P(t) equations require the derivative of the indentation depth 0 8 R i=1 with respect to load. Since experimental data tends to N − P(t) be scattered, the computation of the derivative dH/dP ˙ ˙ τ − J (P τ ) 1 − e P0 i . (3.104) is prone to induce undesirable errors. An alternative i 0 i i=1 approach to determine the creep function under ramp loading is, therefore, described next. Upon least-square fitting (3.104) to the experimentally The representation of the creep function based on measured load–displacement curve one finds the set of , ,..., τ ,τ ,...,τ the generalized Kelvin model is parameters J0 J1 JN and 1 2 N (see again Sect. 3.2.5 for the choice of τ1,τ2,...,τN ), which de- N fine the creep compliance. −t/τi A Part J(t) = J0 + Ji (1 − e ) , (3.100) i=1 Ramp Loading Histories. As noted in Sect. 3.2.6,an ideal step load history cannot be generated in laboratory ... τ τ where J0, J1, , JN are compliance values, 1, 2, tests. Instead one typically uses a ramp loading with 3.3 ...,τ are the retardation times, and N is a positive N a very short rise time t0 first (usually t0 is on the or- integer. Substituting this into (3.94) for the conical in- der of 1–2 s) and a constant load thereafter (Fig. 3.4). denter results in This observation applies equally to large test specimens N (see Sect. 3.2.6) and to nanoindentations. Following 2 1 ˙ then the developments in Sect. 3.2.6,(3.92)and(3.93) H (t) = π(1 − nu)P tan α J + Ji t 4 0 0 i=1 may be used to determine the creep function starting N from a certain time after the constant load is reached. − t τ However, this period of time, which is conservatively − Ji τi 1 − e i . (3.101) chosen as five to ten times the rise time, can be a signif- i=1 icant portion if the total time scale available is not large. ˙ For P(t) = P0 · t,(3.101) can be rewritten as To avoid or at least minimize the loss of the corres- ponding data, one can correct the initial portion of the N data to find the creep compliance between the test start- 2 1 H (t) = π(1 − ν)tanα J + Ji P(t) 4 0 ing time and ten times the rise time using the approach i=1 proposed by Lee and Knauss [3.32]orbyFlory and N − P(t) McKenna [3.33]. P˙ τ − Ji (ν0τi ) 1 − e 0 i . (3.102) Based on the Boltzmann superposition principle i=1 and with reference to Fig. 3.4, a realistic loading can be considered as P(t) = P (t) − P (t) = P˙ th(t) − If the experimentally measured nanoindentation load– 1 2 0 P˙ (t − t )h(t − t ), where P˙ is again a constant load- displacement curve is (least-squares) fitted to (3.102), 0 0 0 0 ing/unloading rate. For a conical indenter, from (3.90), a set of best-fit parameters J , J ,...,J and 0 1 N we have τ1,τ2,...,τN can be determined (Sect. 3.2.5 with re- spect to the choice of τ ,τ ,...,τ ). These constants t 1 2 N π(1 − ν)P˙ tan α define the shear creep compliance (3.100). H2(t) = 0 J(ξ)dξ ; (t < t ) , 4 0 The same method for data reduction can be applied 0 to a spherical indenter. With the substitution of (3.100) (3.105) into (3.91)thisleadsto π − ν ˙ α 2 = (1 )P0 tan − ν ˙ N H (t) 3/2 3(1 )P0 4 H (t) = √ J0 + Ji t t t−t0 8 R = i 1 ξ ξ − ξ ξ ; ≥ . N × J( )d J( )d (t t0) t − τ − Ji τi 1 − e i . (3.103) 0 0 i=1 (3.106) 80 Part A Solid Mechanics Topics Differentiation of (3.105)and(3.106) with respect to than ≈ 0.5%, it is expected that nonlinearly viscoelas- time t yields tic deformations arise. Linearly viscoelastic analysis 8H(t) dH(t) should thus be considered a first-order approximation J(t) = ; (t < t0) , (3.107) for measuring linearly viscoelastic functions. Indeed, π(1 − ν)P˙ tan α dt 0 the linearly viscoelastic analysis has been shown to 8H(t) dH(t) be a good approximation under a variety of situations. J(t − t ) = J(t) − ; For example, Cheng et al. [3.83] have determined that 0 π − ν ˙ α (1 )P0 tan dt the standard linear solid model can be appropriate for ≥ . (t t0) (3.108) some polymers if a sufficiently small time range is Similarly, for a spherical indenter, the following results involved. Hutcheson and McKenna [3.92] found that are obtained: linearly viscoelastic analysis is applicable to the em- √ 1/2 bedment of nanospheres into a polystyrene surface as = 4 RH (t) dH(t) ; < , demonstrated on data obtained by Teichroeb;andFor- atA Part J(t) ˙ (t t0) (3.109) (1 − ν)P0 dt rest [3.93]andOyen [3.94] have demonstrated that √ linearly viscoelastic analysis is appropriate for at least RH1/2 t H t some materials under spherical nanoindentation. On the 3.3 − = − 4 ( ) d ( ) ; ≥ . J(t t0) J(t) (t t0) other hand, others have found that linearly viscoelas- (1 − ν)P˙ dt 0 tic analysis is not applicable to some materials or under (3.110) particular conditions: Thus, in an early application of Therefore, the procedure of data correction could be indentation to viscoelastic properties determination, Va- considered as backward recursion starting at some time, landingham et al. [3.95] found for several polymers for example, ten times the rise time t0. For a conical that the relaxation modulus as determined from dif- indenter, using (3.107)and(3.108), the creep function ferently sized step displacements depended on their determined by (3.92) can be corrected through the fol- magnitude. Since linearly viscoelastic functions are lowing steps: considered to be properties – i. e., they should be inde- pendent of the stress and of the deformation amplitude 1. For kt ≤ t ≤ (k + 1)t with k ≥ 10 being a pos- 0 0 – in nano/microindentation measurements this would itive integer, compute J(t − t )att = kt + mλt 0 0 0 seem to be an indication that nonlinear response is by (3.108); the result of J(t) is calculated using involved. If interest rests on the linearly viscoelastic (3.92), where λ is some sufficiently small number, functions one should ensure that the measurement re- m is an increasing integer from 1, and 0 < m ≤ 1/λ. sults are independent of load or deformation level. Note that λt is the time increment used in backward 0 A comment is in order with respect to the short- recursion. term or glassy response in such measurements. It is 2. For (k −1)t ≤ t ≤ kt , compute J(t −t )att = (k − 0 0 0 noted from (3.96)and(3.99) that the instantaneous 1)t + mλt by (3.108) and the result from (1). 0 0 shear creep compliance is zero at time t = 0 because 3. Repeat the same step as (2) for (n − 1)t ≤ t ≤ nt , 0 0 h(0) = 0 under loading at a constant load rate. Polymers where n = k − 1, k − 2,...,3. normally have nonzero instantaneous creep compliance. 4. For 0 ≤ t < t compute J(t)by(3.107). 0 The error on the instantaneous creep compliance is the For a spherical indenter, the same method can be result of the limitation in the viscoelastic analysis since used to augment the initial part of creep function. the solution is singular due to the sharp point disconti- Simply replace (3.92), (3.107), and (3.108) in (1)–(4) nuity of the tip at zero indentation depth. It is found, for a conical indenter by (3.93), (3.109), and (3.110), however, that after passing the initial loading stage, respectively. the creep compliance typically increases with time, and approaches the value representing the viscoelastic be- Limitations of Micro/Nanoindentation in the Deter- havior. For the initial contact in nanoindentation, the mination of Linearly Viscoelastic Functions. Nanoin- solution details of the viscoelastic problem are rather dentation uses sharp, pointed indenters to penetrate complex, involving the effects of (molecular) repulsion, into a test specimen, and leads to relatively large adhesion, and friction, as well as initial plowing through deformations under the indenter tip, especially when the material. An analysis that takes into account all of a Berkovich indenter is used. Since viscoelastic ma- these factors might be necessary to develop a method terials often exhibit nonlinear behavior at strains larger to determine the instantaneous creep compliance. How- Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 81 ever, experience at a much larger length scale tells us point below). Loubet et al. [3.96] presented the follow- that such an expectation is not well placed. Also, from ing equations to compute the complex modulus E∗(ω) a purely operational point of view in consideration of √ the fact that nanoindenters cannot provide accurate in- ∗ πS E (ω) = E + iE , with E = √ formation for very small depths (of the order of 50 nm 2 A or less) data for the creep function at short times are √ πCω usually not very accurate. and E = √ , (3.111) 2 A Specimen Preparation. The method(s) described above for nanoindentation/microindentation on polymers as- where E and E are the uniaxial storage modulus and sumes that the material is in its natural, stress-free state the loss modulus, respectively. S is a contact stiffness as the reference configuration. Specimens need to be defined as the local slope of the relation between the / prepared carefully prior to the start of measurements. load and the penetration depth, dP dH, C a damping atA Part They typically need to be annealed at a temperature coefficient defined through the instrument software as = / ˙ in a range of ±10 ◦C of the glass-transition tempera- the ratio of the load to the penetration rate, C P H, ture for 2 h or longer to remove any residual stress; and A the contact area between the indenter and the they then need to be cooled slowly (typical cooling workpiece as determined from the penetration depth and 3.3 rates ≈ 5 ◦C/min) to room or the test temperature. The the indenter geometry. This method was employed in physical aging time must be maintained at the same the quoted reference [3.96], for example, to measure the value for all tests to produce consistent results, unless complex modulus of polyisoprene. of course the effect of physical aging time is under As this work took no particular note of the is- study. The room temperature has to be recorded, as well sues associated with the viscoelastic effects resulting as the room humidity, which needs to be controlled from decreasing contact during part of the load cycle, to a constant value using a humidifier/dehumidifier the method is suspect. Therefore, Huang et al. [3.85] if the temperature control unit does not offer this conducted measurements, using also an MTS Nano capability. Indenter XP system, which was also equipped with a continuous stiffness module, but with the specific Measurements of Viscoelastic Functions intent of elucidating the need to bring viscoelasticity in the Frequency Domain theory in accord with the test conditions (contact reten- To illustrate the current state of development with tion). This was accomplished for the spherical indenter nanoindentation equipment and data interpretation we showninFig.3.27b. review here briefly earlier studies that suffer from an in- Leaving the details of development to the reader’s adequate attention to the details of viscoelastic vis-à-vis individual study, we go to the heart of the matter by elastic analysis. Loubet et al. [3.96] proposed a method pointing out that experimental provisions need to be to determine the complex modulus of viscoelastic ma- made to prevent the occurrence of reduction in con- terials with the aid of an MTS Nano Indenter XP system tact between the indenter and the substrate. Huang et al. coupled with a continuous stiffness module (CSM). provided for this by imposing a preload (carrier load) The CSM allows cyclic excitation in load or displace- onto which a much smaller harmonic load was super- ment and the recording of the resulting displacement or posed. This is accomplished through either a constant load [3.97]. The indentation displacement response and (creep) load or through a load that increases at a con- the out-of-phase angle between the applied harmonic stant rate. We record first the results for the constant force and the corresponding harmonic displacement are carrier load and then for the constant carrier load rate, measured continuously at a given excitation frequency. after which the conditions for nondecreasing contact For the subsequent discussion the reader is again alerted area are stated. to the fact that currently available indentation solutions Consider then a sinusoidal indentation load super- require non-decreasing contact area between the ma- imposed on a step loading, represented by terial and the indenter. When that condition cannot be guaranteed, the measurement results must be considered P(t) = Pm + h(t)ΔP0 sin ωt , (3.112) suspect and usually require careful examination and evaluation (see the earlier caveat-discussion on tran- where Pm is the (constant) carrier or main load,and sientsinSect.3.2.6 as well as further discussion on this ΔP0 is the amplitude of the harmonic load. Inserting 82 Part A Solid Mechanics Topics (3.112)into(3.91), we have Following similar procedures as in deriving (3.119), − ν the formulae to determine the complex compliance can 3/2 = 3(1√ ) H (t) also be derived under the condition that the time t has 8 R Δ t evolved to a value such that Hm(t) H0. Substituting (3.119)into(3.91) for the spherical indenter, one has + ωΔ − θ ωθ θ . × Pm J(t) P0 J(t )cos d then 0 t (3.113) − ν 3/2 3(1 ) ˙ √ H (t) = √ P0 J(t − θ)dθ The contact radius is a(t) = RH(t)forH(t) R. 8 R After the loading transients have died out (see the dis- 0 cussion in Sect. 3.2.6), one finds + ΔP [J (ω) sin ωt − J (ω)cosωt] . − ν 0 3/2 3(1 ) H (t) = √ Pm J(t) atA Part (3.120) 8 R + ΔP0[J (ω) sin ωt − J (ω)cosωt] . Upon comparing (3.120) with (3.116), the same for- (3.114) mulasasin(3.118) for the complex compliance can be 3.3 with J (ω)andJ (ω) denoting the storage and loss derived for a small oscillatory load that is superimposed upon a constant-rate carrier load. compliances in shear, respectively. If Hm(t) denotes the We next provide conditions on the load magni- carrier displacement component, and ΔH0 is the am- plitude of the harmonic displacement component, the tude(s) under which nondecreasing contact area is displacement from (3.112)isintheform maintained so that the solution derived from the Lee– Radok approach is valid. These conditions are sufficient = + Δ ω − δ , H(t) Hm(t) H0 sin( t ) (3.115) because they are imposed such that the total load rate where δ is the phase angle between the harmonic force does not become negative, although it is conceivable and the ensuing displacement, ΔH0 is of the order of that a small negative load rate does not necessarily lead a few nanometers while Hm(t) (from the step loading) to a reduction in contact area. Note that similar argu- is on the order of a few hundreds of nanometers. As- mentation cannot be used if the prescribed loading is in suming that this implies that no loss in contact occurs the form of displacement histories. and that ΔH0 Hm(t), (3.115) leads to For a harmonic loading superimposed on a step loading one ensures (small) positive loading rates by re- 3/2 3/2 3 1/2 H (t) = Hm (t) + Hm (t)ΔH0 cos δ sin ωt quiring that, for each arbitrary time interval, say half 2 a harmonic cycle, the indentation rate due to the con- 3 1/2 − H (t)ΔH sin δ cos ωt + o(ΔH ) , stant carrier load exceeds temporary unloading. From 2 m 0 0 (3.116) (3.115) the contact area will be nondecreasing during the whole process, as long as the frequency does not where o(ΔH0) indicates the higher-order terms in ΔH0, ˙ exceed the critical value ωc = Hm/ΔH0.Avaluefor which are negligible as long as ΔH0 Hm(t). Com- ˙ Hm may be estimated from (3.117) as if the relation paring (3.116) with (3.114), one finds for the constant described an elastic half-space. For higher frequencies carrier load that a temporary decrease in the contact is likely as a result 3/2 3(1 − ν) Hm (t) = √ Pm J(t) , (3.117) of the applied harmonic load so that the Ting approach 8 R should be adopted. Nevertheless, when the frequency √ / exceeds the critical value by a small amount (ω>ωc), 4 R H1 2(t)ΔH J ω = m 0 δ , the solutions derived from the methods by Lee and ( ) − ν Δ cos and 1 √ P0 Radok [3.87] and by Ting [3.86] are still very close, 1/2 even though the condition for the Lee–Radok approach 4 R Hm (t)ΔH0 J (ω) = sin δ . (3.118) is not strictly fulfilled. Since a closed-form solution de- 1 − ν ΔP 0 rived from the Lee–Radok approach exists, while only The alternative loading condition in which a small numerical solutions can be obtained using the Ting sinusoidal load is superimposed upon a carrier loading ˙ approach, the formulas derived for a harmonic superim- increasing at a constant rate P0,i.e., posed on a step loading from the Lee–Radok approach ˙ P(t) = P0t + h(t)ΔP0 sin ωt . (3.119) can then still be used to estimate the complex viscoelas- Mechanics of Polymers: Viscoelasticity 3.3 Measurements and Methods 83 tic functions in the regime of linear viscoelasticity when value, indicating considerable uncertainty associated ω>ωc. with the method by Loubet et al. [3.96] as summarized Next consider the carrier load to increase at a con- by (3.111) for measuring the storage modulus. This dis- ˙ stant rate P0. Differentiation of (3.119) with respect crepancy exists for both PC and for PMMA. to time guarantees a positive loading rate as long as ˙ P0 ≥ ΔP0ω. Additionally, the substitution√ of this in- 3.3.5 Photoviscoelasticity equality into (3.91) together with a(t) = RH(t)shows that the contact area will then also not decrease during A classical tool for determining strain and stress the entire indentation history. distributions in two-dimensional geometries is the Figure 3.28 shows a comparison of these develop- photoelastic method [3.98]. For three-dimensional ge- ments with an application of (3.111)[3.96] under the ometries the method required slicing the body into assumption of a constant Poisson ratio: when meas- sections and treat each two-dimensional section/slice urements over a relatively short time (such as ≈ 250 s separately and sequentially. While the early appli- atA Part used in this study) are made, the Poisson’s ratio [3.15] cation of this technique employed relatively rigid does not change significantly for polymers in the glassy polymers such as homalite (a polyester) or polystyrene state, such as PMMA or polycarbonate (PC) and thus with glass temperatures above 100 ◦C, also softer introduces negligible errors in the complex compliance and photoelastically more sensitive materials such as 3.3 data. To compute the complex modulus of PC and polyurethane elastomers have been employed [3.99]. polymethyl methacrylate (PMMA) at 75 Hz, data were This method was found useful in both quasistatic as acquired continuously at this frequency for ≈ 125 s. well as dynamic applications when wave mechanics In general the data increase correctly with time, and was an important consideration [3.100]. With respect approach a nearly constant value for each material. to viscoelastic responses it is important to consider These constant values are considered to represent the the time scale of the measurements relative to time steady state and are quoted as the storage modulus. Also of the test material. Rigid polymers are stiff be- shown for comparison in Fig. 3.28 are data measured cause their dominant relaxation processes occur slowly with the aid of conventional dynamic mechanical ana- around typical laboratory temperatures, so that time- lysis (DMA) for the same batch of PC and PMMA. dependent issues are not of much concern. On the The uniaxial storage modulus of PC measured by DMA other hand, they should be of concern if observa- at 0.75 Hz is 2.29 GPa. However, the storage modulus tions extend over long time periods measured in weeks computed using (3.111) is at least 40% higher than this and months if the relaxation times at the prevail- a) Storage modulus (GPa) b) Storage modulus (GPa) 8 8 7 7 6 6 5 5 4 4 3 3 Nanoindentation (2004) Nanoindentation (2004) 2 Conventional (DMA) 2 Conventional (DMA) Nanoindentation (1995) Nanoindentation (1995) 1 1 0 0 02550 75 100 125 150 02550 75 100 125 150 Time (s) Time (s) Fig. 3.28a,b Comparison of the storage compliance at 75 Hz computed by three methods for (a) PC and (b) PMMA 84 Part A Solid Mechanics Topics ing temperatures are of the same order of magnitude. sition to purely elastic behavior for the relatively very Because dynamic events occur in still shorter time long times) are likely to be excited so that the output of frames wave mechanics typically little concern in this the measurements must consider the effect of viscoelas- regard. tic response. The situation is quite different when soft or elas- It is beyond the scope of this presentation to de- tomeric polymers serve as photoelastic model materials. lineate the full details of the use of viscoelastically In that case quasistatic environments (around room photoelastic material behavior, especially since during temperature) typically involve only the long-term or the past few years investigators have shown a strong rubbery behavior of the material with the stiffness meas- inclination to use alternative tools. However, it seems ured in terms of the rubbery or long-term equilibrium useful to include a list of references from which the modulus. On the other hand, when wave propagation evolution of this topic as well as its current status may phenomena are part of the investigation, the longest be explored. These are listed as a separate group in relaxation times (relaxation times that govern the tran- Sect. 3.6.1 under References on Photoviscoelasticity. atA Part 3.4 3.4 Nonlinearly Viscoelastic Material Characterization Viscoelastic materials are often employed under con- the multiaxial stress state. References regarding non- ditions fostering nonlinear behavior. In contrast to the linear behavior in the context of uniaxial deformations mutual independence in the dilatational and deviatoric are too numerous to list here. The reader is advised responses in a linearly viscoelastic material, the vis- to consult the following journals: the Journal of Poly- coelastic responses in different directions are coupled mer, Applied Polymer Science, Polymer Engineering and must be investigated in multiaxial loading con- and Science, the Journal of Materials Science and Me- ditions. Most results published in the literature are chanics of Time-Dependent Materials, to list the most restricted to investigating the viscoelastic behavior in prevalent ones. the uniaxial stress or uniaxial strain states [3.101, 102] and few results are reported for time-dependent 3.4.1 Visual Assessment multiaxial behavior [3.103–105]. Note that Bauwens- of Nonlinear Behavior Crowet’s study [3.104] incorporates the effect of pressure on the viscoelastic behavior but not in the Although it is clear that even under small deformations data analysis, simply as a result of using uniaxial entailing linearly viscoelastic behavior the imposition compression deformations. Along similar lines Knauss, of a constant strain rate on a tensile or compression Emri, and collaborators [3.106–110] provided a se- specimen results in a stress response that is not linearly ries of studies deriving nonlinear viscoelastic behavior related to the deformation when the relation is estab- from changes in the dilatation (free-volume change) lished in real time (not on a logarithmic time scale). which correlated well with experiments and gave at Because such responses that appear nonlinear on paper least a partial physical interpretation to the Schapery are not necessarily indicative of nonlinear constitutive scheme [3.111, 112] for shifting linearly viscoelastic behavior, it warrants a brief exposition of how nonlin- data in accordance with a stress or strain state. The ear behavior is unequivocally separated from the linear studies became the precursors to investigate the effect type. of shear stresses or strains on nonlinear behavior as To demonstrate the nonlinear behavior of a ma- described below. Thus the proper interpretation of the terial, consider first isochronal behavior of a linearly uniaxial data as well as their generalization to multiax- viscoelastic material. Although isochronal behavior can ial stress or deformation states is highly questionable. be obtained for different deformation or stress histories, This remains true regardless of the fact that such data consider the case of shear creep under various stress lev- interpretation has been incorporated into commercially els σn, so that the corresponding strain is εn = J(t)σn available computer codes. Certainly, there are very few for any time t. Consider an arbitrary but fixed time t∗, engineering situations where structural material use is at which time the ratio limited to uniaxial states. In this section we describe ∗ some aspects of nonlinearly viscoelastic behavior in εn/σn = J(t ) (3.121) Mechanics of Polymers: Viscoelasticity 3.4 Nonlinearly Viscoelastic Material Characterization 85 takes on a particular property value. At that time all pos- ial. These values are likely to be different for other sible values of σn and εn are related linearly: a plot materials. of σ versus ε at time t∗ renders a linear relation with slope 1/J(t∗)andencompassing the origin. For differ- 3.4.2 Characterization ent times t∗, straight lines with different slopes result of Nonlinearly Viscoelastic Behavior and the linearly viscoelastic material can thus be char- Under Biaxial Stress States acterized by a fan of straight lines emanating from the origin, the slope of each line corresponding to a dif- In the following sections, we describe several ways of ferent time t∗. The slopes decrease monotonically as measuring nonlinearly viscoelastic behavior in multiax- these times increase. Each straight line is called a lin- ial situations. ear isochronal stressÐstrain relation for the particular time t∗. Hollow Cylinder under Axial/Torsional Loading Figure 3.29 shows such an isochronal representa- A hollow cylinder under axial/torsional loading condi- tion for PMMA. It is seen that, at short times and when tions provides a vehicle for investigating the nonlinearly A Part the strains are small in creep (shear strain 0.005, shear viscoelastic behavior under multiaxial loading con- stress 8 MPa), the material response is close to linearly ditions. Figure 3.30 shows a schematic diagram of viscoelastic where all the data stay within the linear fan a cylinder specimen. Dimensions can be prescribed 3.4 formed, in this case, by the upper line derived from corresponding to the load and deformation range of the shear creep compliance at 10 s, and the lower line interest as allowed by an axial/torsional buckling corresponding to the compliance in shear at 104 s. At analysis. the higher stress levels and at longer times, when the Two examples are given here for specimen dimen- strains are larger, material nonlinearity becomes pro- sions. With the use of a specimen with outer diameter nounced, as data points in Fig. 3.29 are outside the of 22.23 mm, a wall thickness of 1.59 mm, and a test linear fan emanating from the origin. In this isochronal length of 88.9 mm, the ratio of the wall thickness to plot, the deviation from linearly viscoelastic behavior the radius is 0.14. These specimens can reach a surface begins at approximately 0.5% strain level. Isochronal shear strain on the order of 4.0–4.5%. With the use of data at other temperatures indicate also that the non- an outer diameter of 25.15 mm, a thickness of 3.18 mm, linearity occurs at ≈ 0.5% strain and a shear stress and a test length of 76.2 mm, the ratio of the wall thick- of about 7.6 MPa at all temperatures for this mater- ness to the radius is 0.29. This allows a maximum shear strain prior to buckling in the range of 8.5–12.5% based on elastic analysis. The actual maximum shear strain Shear stress (MPa) that can be achieved prior to buckling can be slightly 16 different due to the viscoelastic effects involved in the 14 material. Estimates of strains leading to buckling may t = 10s be achieved by considering a Young’s modulus for the 12 material that corresponds to the lowest value achieved t = 104s at the test temperature and the time period of interest 10 for the measurements. 8 x 6 Fig. 3.30 A thin- 4 walled cylinder specimen for 2 D y combined ten- O 0 sion/compression 0 0.01 0.02 0.03 z and torsion to Shear strain generate bi- Fig. 3.29 Isochronal shear stress–shear strain relation of axial stress PMMA at 80 ◦C states [3.113] 86 Part A Solid Mechanics Topics Application of Digital Image Correlation age correlation techniques [3.115, 116], and corrected The use of strain gages for determining surface strains for curvature to determine the axial, circumferential, on a polymer specimen is fraught with problems, since and shear strains [3.114]. To assure that the motion is strain gages tend to be much stiffer than the polymer properly interpreted in a cylindrical coordinate system undergoing time-dependent deformations, and, in ad- – that the camera axis is effectively very well aligned dition, the potential for increases in local temperature with the cylinder axis and test frame orientation – the due to the currents in the strain gage complicates defini- imaging system must also establish the axis of rotation. tive data evaluation. Digital image correlation (DIC, This can be achieved through offsetting the specimen Chap. 20) thus offers a perfect tool, though that method against a darker background (Fig. 3.31)soastoen- is not directly applicable to cylindrical surfaces as typ- sure sufficient contrast between the specimen edge and ically employed. While we abstain from a detailed background for identification of any inclination of the review of this method in this context (we refer the reader cylinder axis relative to the reference axis within the to [3.114] for particulars), here it is of interest for the image recording system. The axis orientation is then atA Part completeness of presentation only to summarize this also evaluated using the principles of digital image cor- special application to cylindrical surface applications. relation. Without such a determination the parameters The results are presented in the form of (apparent) creep identifying the projection of the cylindrical surface onto 3.4 compliances defined by 2ε(t)/τ0. We emphasize again a plane lead to uncontrollable errors in the data interpre- that the creep compliance for a linear material is a func- tation. The details of the relevant data manipulation can tion that depends only on time but not on the applied be found in the cited references. stress. This no longer holds in the nonlinearly viscoelas- tic regime, but we adhere to the use of this ratio as Specimen Preparation a creep compliance for convenience. Specimens can be machined from solid cylinders or To use DIC for tracking axial, circumferential, and from tubes, though tubular specimens tend to have shear strains on a cylindrical surface, a speckle pattern a different molecular orientation because of the extru- is projected onto the specimen surface. While the same sion process. Prior to machining, the cylinders need to image acquisition system can be used as for flat images be annealed at a temperature near the glass-transition special allowance needs to be made for the motion of temperature to remove residual stresses. The thin- surface speckles on a cylindrical surface, the orienta- walled cylinder samples need to be annealed again after tion of which is also not known a priori. If the focal machining to remove or reduce residual surface stresses length of the imaging device is long compared to the possibly acquired during turning. To avoid excessive radius of the cylinder, an image can be considered as gravity deformations, annealing is best conducted in a projection of a cylinder onto an observation plane. an oil bath. Any possible weight gain must be mon- Planar deformations can be determined using digital im- itored with a balance possessing sufficient resolution. The weight gain should be low enough to avoid any effect of the oil on the viscoelastic behavior of mater- ials. For testing in the glassy state, specimens must have about the same aging times; and the aging times should be at least a few days so that during measurements the aging time change is not significant (on a logarith- mic scale). Because physical aging is such an important topic we devote further comments to it in the next sub- section. Prior to experiments samples need to be kept in an environment with a constant relative humidity that is the same as the relative humidity during measurements. The relative humidity can be generated through a satu- rated salt solution in an enclosed container [3.118]. Physical Aging in Specimen Preparation. When an Fig. 3.31 A typical speckle pattern on a cylinder surface amorphous polymer is cooled (continuously) from its inclined with respect to the observation axis of the imaging melt state, its volume will deviate from its equilib- system rium state at the glass-transition temperature (Fig. 3.12). Mechanics of Polymers: Viscoelasticity 3.4 Nonlinearly Viscoelastic Material Characterization 87 Polymers have different viscoelastic characteristics de- the time required to attain equilibrium after quench- pending on whether they are below the glass-transition ing within practical limits. The value of t∗ may have temperature (Tg), in the glass-transition region (in the to be determined prior to commencing characteriza- neighborhood of Tg), or in the rubbery state (above tion tests. If the viscoelastic properties are investigated Tg). In the rubbery state a polymer is in or near ther- without paying attention to the aging process, character- modynamic equilibrium, where long-range cooperative ization of polymers and their composites are not likely motions of long-chain molecules are dominant and re- to generate repeatable results. For characterization of sult in translational movements of molecules. Below the the long-term viscoelastic behavior through accelerated glass-transition temperature, short-range motions in the testing, physical aging effects have to be considered, form of side-chain motions and rotations of segments in addition to time–temperature superposition and other of the main chain (primarily in long-term behavior) mechanisms. are dominant. The glass-transition range depends on the cooling rate. After cooling a polymer initially in An Example of Nonlinearly Viscoelastic atA Part the rubbery state to an isothermal condition in the Behavior under Combined Axial/Shear Stresses glassy state, the polymer enters a thermodynamically Figure 3.32 shows the creep response in pure shear ◦ nonequilibrium, or metastable, state associated with for PMMA at 80 C[3.113, 117]. The axial force was a smaller density than an optimal condition (equilib- controlled to be zero in these measurements. The plot- 3.4 rium) would allow. In the equilibrium state the density ted shear creep compliance was converted from the would increase continuously to its maximum value. If relaxation modulus in shear under infinitesimal defor- the temperature in the isothermal condition is near Tg, mation, representing the creep behavior in the linearly the density increase can occur in a relatively short time, viscoelastic regime. For a material that behaves lin- but if the temperature is far below Tg this process occurs early viscoelastically, all curves should be coincident in over a long period of time, on the order of days, weeks, this plot. In the case of data at these stress levels, the or months. Prior to reaching the maximum density, deformations in shear at higher stress levels are acceler- as time evolves, depending on how long this process ated relative to the behavior at infinitesimal strains. This has taken place, the polymer possesses a different vis- observation constitutes another criterion for separating coelastic response. This phenomenon is called physical linear from nonlinear response. aging because no chemical changes occur. The time af- To represent the nonlinear characteristics we draw ter quenching to an isothermal condition in the glassy on the isochronal representation discussed above. At state is called aging time. any given time spanned in Fig. 3.32, there are five data The viscoelastic functions (e.g., bulk and shear re- points from creep under a pure shear stress, giving four laxation moduli) change during aging until that process sets of isochronal stress/strain data. Plotting these four is complete within practical time limitations. The effect of physical aging is similar to a continual decrease of the temperature and results in the reduction of the free log(shear creep compliance) (1/MPa) –2.2 volume that provides the space for the mobility of the σ = 0, τ = 16 MPa T = 80°C polymer chain segments as the chain undergoes any re- σ τ –2.4 = 0, = 14.7 MPa arrangement. There now exists a relatively large body σ = 0, τ = 12.3 MPa of information on physical aging and the reader is re- σ = 0, τ = 9.4 MPa ferred to a number of representative publications, in –2.6 which references in the open literature expand on this topic. References [3.38–44,119–122] showed that phys- –2.8 ical aging leads to an aging time factor multiplying the external time, analogous to the temperature-dependent Inversion from µ(t) multiplier (shift factor) for thermorheologically simple –3 solids in the context of linear viscoelasticity theory. Elaborations of this theme for various materials have –3.2 been offered to a large extent by McKenna and by Gates 0.5 1 1.5 2 2.5 3 3.5 4 4.5 as well as their various collaborators log(time) (s) Effects of physical aging can be pronounced be- Fig. 3.32 Shear creep compliance of PMMA at several lev- ∗ fore aging time reaches the value, say, t ,whichis els of shear stress at 80 ◦C(after[3.117]) 88 Part A Solid Mechanics Topics log(axial creep compliance) (1/MPa) A–A Clamped with –2.2 bolt A–A –2.4 Glued for θ > 40° Tension + torsion T σ = 25.3 MPa, τ = 14.5 MPa = 50°C –2.6 θ –2.8 Compression + torsion A σ = 25.3 MPa, τ = 14.5 MPa x –3 y atA Part A –3.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 log(time) (s) 3.4 Fig. 3.33 Axial creep compliance of PMMA under ten- sion/torsion and compression/torsion at 50 ◦C data points at each of the 16 fixed times, say, gives the isochronal stress–strain relation shown in Fig. 3.29. It is clear that the creep rate increases with an in- crease in applied shear stress, indicating nonlinear creep Fig. 3.35 Fixture for testing Arcan specimens behavior in shear. We note that for isochronal behavior at strains ated measurably. It is of interest to note that the above 0.5%, there exists a fan emanating from the creep process is more pronounced (accelerated) in shear strain 0.5% and a shear stress of 7.6MPa. tension/torsion than under compression/torsion as illus- The corresponding fan center is considered to be the trated in Fig. 3.33 for 50 ◦C. We have already observed yield point, above which the creep rate is acceler- that thin-walled cylinders tend to buckle under suf- ficiently high torsion and/or compression. A cylinder with an outer diameter of 25.15 mm, a thickness of Test section 3.18 mm, and a test length of 76.2 mm would buckle 10.16 at ≈ 5% shear strain under pure torsion. The use of R thicker-walled cylinders would reduce the homogene- = 10.16 22.54 ity of the stress and strain within the cylinder wall and lead to inaccuracy in the determination of stress 22.3 or strain. Other techniques, such as testing with the Arcan specimen should, therefore, be used when the nonlinearly viscoelastic behavior at larger deformations 44.6 22.54 is investigated. Use of the Arcan Specimen Arcan’s specimen [3.123–125] can be used for multi- biaxial test with the use of a uniaxial material test system. Figure 3.34 shows an Arcan specimen, and Fig. 3.35 a corresponding test fixture. The loading axis 15.14 can form different angles with respect to the speci- 30.48 men axis so that biaxial stress states can be generated in the region of uniform deformation in the middle of Fig. 3.34 Geometry of an Arcan specimen (all dimensions the specimen. When the loading axis of the fixture is are in mm, thickness is 3 mm) aligned with the major specimen axis, this configura- Mechanics of Polymers: Viscoelasticity 3.5 Closing Remarks 89 Fig. 3.36 Isochronal contours of creep strains under fixed Normal strain biaxial loading. Each contour corresponds to a different 0.025 time between 10 s and about 105 s. Ellipses correspond to linear response characteristics 0.02 0.015 tion induces shear forces applied to an Arcan specimen 0.01 so that there is a pure shear zone in the central portion of 0.005 the specimen. Other orientations allow the nonlinearly viscoelastic shear behavior to be characterized under 0 loading conditions combining tension/shear, compres- –0.005 sion/shear, and pure shear. –0.01 The data processing is illustrated using the results obtained by Knauss and Zhu [3.126,127]asanexample. –0.015 atA Part Figure 3.36 shows isochronal creep shear and normal –0.02 ◦ strains at 80 C using an Arcan specimen under a nomi- –0.025 nal (maximum) shear stress of 19.3 MPa. At each fixed –0.03 –0.02 –0.01 0 0.01 0.02 0.03 time, line segments connect points to form an isochronal Normal strain 3.5 strain contour. The innermost contour corresponds to a creep time of 10 s, and the outermost contour is the re- show ellipses (a/b = 2) that would correspond to totally sults from 0.8×105 s. For comparison purposes we also linearly viscoelastic behavior. 3.5 Closing Remarks As was stated at the very beginning, today’s labora- time scales follow an initially nonlinear deformation tory and general engineering environment is bound to history. It is becoming clear already that the superpo- involve polymers, whether of the rigid or the soft vari- sition of dilatational stresses or volumetric strains has ety. The difference between these two derives merely a greater influence on nonlinear material response of from the value of their glass-transition temperature polymers than is true for metals. Consequently it would relative to the use temperature (usually room tempera- seem questionable whether uniaxial tensile or compres- ture). As illustrated in this chapter the linearized theory sive behavior would be a suitable method for assessing of viscoelasticity is well understood and formulated nonlinear polymer response, since that stress state in- mathematically, even though its current application in volves both shear and bulk (volumetric) components. engineering designs is usually not on a par with this To support this observation one only needs to recall that understanding. A considerable degree of response es- very small amounts of volume change can have a highly timation can be achieved with this knowledge, but disproportionate effect on the time dependence of the a serious deficiency arises from the fact that when struc- material, as delineated in Sects. 3.2.8, 3.2.9,and3.2.10. tural failures are of concern the linearized theory soon This is well illustrated by the best known and large ef- encounters limitations as nonlinear behavior is encoun- fect which a change in temperature has on the relaxation tered. times, where dilatational strains are indeed very small There is, today, no counterpart nonlinear viscoelas- compared to typical shear deformations; responses un- tic material description that parallels the plasticity der pressure and with solvent swelling underscore this theory for metallic solids. Because the atomic structures observation. of metals and polymers are fundamentally different, it The recent publication history for time-dependent would seem imprudent to characterize polymer nonlin- material behavior exhibits an increasing number of ear behavior along similar lines of physical reasoning papers dealing with nonlinear polymer behavior, indi- and mathematical formulation, notwithstanding the fact cating that efforts are underway to address this lack that in uniaxial deformations permanent deformations of understanding in the engineering profession. At the in metals and polymers may appear to be similar. That same time it is also becoming clear that the intrin- similarity disappears as soon as temperature or extended sic time-dependent behavior of polymers is closely 90 Part A Solid Mechanics Topics connected to the molecular processes that are well rep- viscoelastic solids with the expectation that this knowl- resented by the linearly viscoelastic characterization of edge provides a necessary if not sufficient background these solids. It is thus not unreasonable, in retrospect, for dealing with future issues that need to be resolved in to have devoted a chapter mostly to describing linearly the laboratory. 3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known Experimental work deals with a variety of situations/test 3.6.1 Further Reading configurations for which boundary value histories are not readily confined to a limited number of cases. For For general background it appears useful to identify the example, when deformations are to be measured by dominant publications to bolster one’s understanding of photoelasticity with the help of a viscoelastic, photo- the theory of viscoelasticity. To that end we summa- atA Part sensitive material in a two-dimensional domain it may rize here first, without comment or reference number, be important to know the local stress state very well, a list of publications in book or paper form, only one if knowing the stress state in a test configuration is of which appears as an explicit reference in the text, 3.6 an important prerequisite for resolving an engineering namely the book by J.D. Ferry as part of the text de- analysis problem. velopment [3.45] with respect to the special topic of There exists a large class of elastic boundary value thermorheologically simple solids. problems for which the distribution of stresses (or strains) turns out to be independent of the material prop- 1. T. Alfrey: Mechanical Behavior of High Polymers erties. In such cases the effect of loading and material (Interscience, New York, 1948) properties is expressed through a material-dependent 2. B. Gross: Mathematical Structure of the Theories factor and a load factor, both multiplying a function(s) of Viscoelasticity (Herrmann, Paris, 1953) (re-issued that depends only on the spatial coordinates governing 1968) the distribution of stress or strain. It then follows that 3. A.V. Tobolsky: Properties and Structure of Poly- for the corresponding viscoelastic solution the distri- mers (Wiley, New York, 1960) bution of stresses is also independent of the material 4. M.E. Gurtin, E. Sternberg: On the linear theory of properties and that the time dependence is formulated viscoelasticity, Arch. Rat. Mech. Anal. 11, 291–356 as the convolution of a material-dependent multiplica- (1962) tive function with the time-dependent load factored out 5. F. Bueche: Physical Properties of Polymers (Inter- from the spatial distribution function(s). science, New York, 1962) In the experimental context beams and plates fall 6. M.L. Williams: The structural analysis of viscoelas- into this category, though even the simple plate config- tic materials, AIAA J. 2, 785–809 (1964) uration can involve Poisson’s ratio in its deformation 7. Flügge (Ed.): Encyclopedia of Physics VIa/3,M.J. field. More important is the class of simply connected Leitman, G.M.C. Fischer: The linear theory of vis- two-dimensional domains for which the in-plane stress coelasticity (Springer, Berlin, Heidelberg, 1973) distribution is independent of the material properties. 8. J.J. Aklonis, W.J. MacKnight: Introduction to Poly- Consider first two-dimensional, quasistatic prob- mer Viscoelasticity (Wiley, New York 1983) lems with (only) traction boundary conditions pre- 9. N.W. Tschoegl: The Phenomenological Theory scribed on a simply connected domain. For such of Linear Viscoelastic Behavior, an Introduction problems the stress distribution of an elastic solid (Springer, Berlin, 1989) throughout the interior is independent of the mater- 10. A. Drozdov: Viscoelastic Structures, Mechanics of ial properties. The same situation prevails for multiply Growth and Aging (Academic, New York, 1998) connected domains, provided the traction on each per- 11. D.R. Bland: The Theory of Linear Viscoelasticity, foration is self-equilibrating. If the latter condition is Int. Ser. Mon. Pure Appl. Math. 10 (Pergamon, New not satisfied, then a history-dependent Poisson function York, 1960) enters the stress field description so that the stress dis- 12. R.M. Christensen: Theory of Viscoelasticity: An In- tribution is, at best, only approximately independent of, troduction (Academic, New York, 1971); see also or insensitive to, the material behavior. This topic is R.M. Christensen: Theory of Viscoelasticity An In- discussed in a slightly more detailed manner in [3.13]. troduction, 2nd ed. (Dover, New York, 1982) Mechanics of Polymers: Viscoelasticity 3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known 91 References on Photoviscoelasticity 17. C.W. Folkes: Two systems for automatic reduc- 1. R.D. Mindlin: A mathematical theory of photovis- tion of time-dependent photomechanics data, Exp. coelasticity, J. Appl. Phys. 29, 206–210 (1949) Mech. 10, 64–71 (1970) 2. R.S. Stein, S. Onogi, D.A. Keedy: The dynamic 18. P.S. Theocaris: Phenomenological analysis of me- birefringence of high polymers, J. Polym. Sci. 57, chanical and optical behaviour of rheo-optically 801–821 (1962) simple materials. In: Photoelastic Effect and its ap- 3. C.W. Ferguson: Analysis of stress-wave propaga- plications, ed. by J. Kestens (Springer, Berlin New tion by photoviscoelastic techniques, J. Soc. Motion York, 1975) pp. 146–152 Pict. Telev. Eng. 73, 782–787 (1964) 19. B.D. Coleman, E.H. Dill: Photoviscoelasticity: The- 4. P.S. Theocaris, D. Mylonas: Viscoelastic effects in ory and practice. In: The Photoelastic Effect and birefringent coating, J. Appl. Mech. 29, 601–607 its Applications, ed. by J. Kestens, (Springer, Berlin (1962) New York, 1975) pp. 455–505 5. M.L. Williams, R.J. Arenz: The engineering ana- 20. M.A. Narbut: On the correspondence between dy- atA Part lysis of linear photoviscoelastic materials, Exp. namic stress states in an elastic body and in its Mech. 4, 249–262 (1964) photoviscoelastic model, Vestn. Leningr. Univ. Ser. 6. E.H. Dill: On phenomenological rheo-optical con- Mat. Mekh. Astron. (USSR) 1, 116–122 (1978) stitutive relations, J. Polym. Sci. Part C 5, 67–74 21. R.J. Arenz, U. Soltész: Time-dependent optical 3.6 (1964) characterization in the photoviscoelastic study of 7. C.L. Amba-Rao: Stress-strain-time-birefringence stress-waver propagation, Exp. Mech. 21, 227–233 relations in photoelastic plastics with creep, J. (1981) Polym. Sci. Pt. C 5, 75–86 (1964) 22. K.S. Kim, K.L. Dickerson, W.G. Knauss (Eds): Vis- 8. B.E. Read: Dynamic birefringence of amorphous coelastic effect on dynamic crack propagation in polymers, J. Polym. Sci. Pt. C 5, 87–100 (1964) Homalit 100. In: Workshop on Dynamic Fracture 9. R.D. Andrews, T.J. Hammack: Temperature depen- (California Institute of Technology, Pasadena, 1983) dence of orientation birefringence of polymers in pp. 205–233 the glassy and rubbery states, J. Polym. Sci. Pt. C 23. H. Weber: Ein nichtlineares Stoffgesetz für die 5, 101–112 (1964) ebene photoviskoelastische Spannungsanalyse, Rheol. 10. R. Yamada, C. Hayashi, S. Onogi, M. Horio: Dy- Acta. 22, 114–122 (1983) namic birefringence of several high polymers, J. 24. Y. Miyano, S. Nakamura, S. Sugimon, T.C. Woo: Polym. Sci. Pt. C 5, 123–127 (1964) A simplified optical method for measuring residual 11. K. Sasguri, R.S. Stain: Dynamic birefringence of stress by rapid cooling in a thermosetting resin strip, polyolefins, J. Polym. Sci. Pt. C 5, 139–152 (1964) Exp. Mech. 26, 185–192 (1986) 12. D.G. Legrand, W.R. Haaf: Rheo-optical properties 25. A. Bakic: Practical execution of photoviscoelas- of polymers, J. Polym. Sci. Pt. C 5, 153–161 (1964) tic experiments, Oesterreichische Ingenieur- und 13. I.M. Daniel: Experimental methods for dynamic Architekten-Zeitschrift, 131, 260–263 (1986) stress analysis in viscoelastic materials, J. Appl. 26. K.S. Kim, K.L. Dickerson, W.G. Knauss: Viscoelas- Mech. 32, 598–606 (1965) tic behavior of opto-mechanical properties and its 14. I.M. Daniel: Quasistatic properties of a photovis- application to viscoelastic fracture studies, Int. J. coelastic material, Exp. Mech. 5, 83–89 (1965) Fract. 12, 265–283 (1987) 15. A.J. Arenz, C.W. Ferguson, M.L. Williams: 27. T. Kunio, Y. Miyano, S. Sugimori: Fundamentals The mechanical and optical characterization of of photoviscoelastic technique for analysis of time a Solithane 113 composition, Exp. Mech. 7, 183– and temperature dependent stress and strain. In: Ap- 188 (1967) plied Stress Analysis, ed. by T.H. Hyde, E. Ollerton 16. H.F. Brinson: Mechanical, optical viscoelastic char- (Elsevier Applied Sciences, London, 1990) pp. 588– acterization of Hysol 4290: Time and temperature 597 behavior of Hysol 4290 as obtained from creep tests 28. K.-H. Laermann, C.Yuhai: On the measurement of in conjunction with the time-temperature superposi- the material response of linear photoviscoelastic tion principle, Exp. Mech. 8, 561–566 (1968) polymers, Measurement, 279–286 (1993) 92 Part A Solid Mechanics Topics 29. S. Yoneyama, J. Gotoh, M. Takashi: Experimental from the glassy to rubbery state, J. Polym. Sci. Pt. B analysis of rolling contact stresses in a viscoelastic Polym. Phys. 39, 2252–2262 (2001) strip, Exp. Mech. 40, 203–210 (2000) 31. Y.-H. Zhao, J. Huang: Photoviscoelastic stress ana- 30. A.I. Shyu, C.T. Isayev, T.I. Li: Photoviscoelastic lysis of a plate with a central hole, Exp. Mech. 41, behavior of amorphous polymers during transition 312–18 (2001) References 3.1 W.G. Knauss: The mechanics of polymer fracture, 3.18 R.A. Schapery: Approximate methods of transform Appl. Mech. Rev. 26,1–17(1973) inversion for viscoelastic stress analysis, Proc. 4th 3.2 C. Singer, E.J. Holmgard, A.R. Hall (Eds.): AHistoryof US Natl. Congr. Appl. Mech. (1962) pp. 1075–1085 Technology (Oxford University Press, New York 1954) 3.19 J.F. Clauser, W.G. Knauss: On the numerical deter- atA Part 3.3 J.M. Kelly: Strain rate sensitivity and yield point be- mination of relaxation and retardation spectra for havior in mild steel, Int. J. Solids Struct. 3,521–532 linearly viscoelastic materials, Trans. Soc. Rheol. 12, (1967) 143–153 (1968) 3 3.4 H.H. Johnson, P.C. Paris: Subcritical flaw growth, 3.20 G.W. Hedstrom, L. Thigpen, B.P. Bonner, P.H. Wor- Eng. Fract. Mech. 1,3–45(1968) ley: Regularization and inverse problems in vis- 3.5 I. Finnie: Stress analysis for creep and creep- coelasticity, J. Appl. Mech. 51, 121–12 (1984) rupture. In: Appllied Mechanics Surveys,ed.by 3.21 I. Emri, N.W. Tschoegl: Generating line spectra from H.N. Abramson (Spartan Macmillan, New York 1966) experimental responses. Part I: Relaxation modu- pp. 373–383 lus and creep compliance, Rheol. Acta. 32, 311–321 3.6 F. Garofalo: Fundamentals of Creep and Creep Rup- (1993) ture in Metals (Macmillan, New York 1965) 3.22 I. Emri, N.W. Tschoegl: Generating line spectra from 3.7 N.J. Grant, A.W. Mullendore (Eds.): Deformation and experimental responses. Part IV: Application to ex- Fracture at Elevated Temperatures (MIT Press, Cam- perimental data, Rheol. Acta. 33, 60–70 (1994) bridge 1965) 3.23 I. Emri, N.W. Tschoegl: An iterative computer al- 3.8 F.A. McClintock, A.S. Argon (Eds.): Mechanical Be- gorithm for generating line spectra from linear havior of Materials (Addison-Wesley, Reading 1966) viscoelastic response functions, Int. J. Polym. Mater. 3.9 J.B. Conway: Numerical Methods for Creep and Rup- 40, 55–79 (1998) ture Analyses (Gordon-Breach, New York 1967) 3.24 N.W. Tschoegl, I. Emri: Generating line spectra from 3.10 J.B. Conway, P.N. Flagella: Creep Rupture Data experimental responses. Part II. Storage and loss for the Refractory Metals at High Temperatures functions, Rheol. Acta. 32, 322–327 (1993) (Gordon-Breach, New York 1971) 3.25 N.W. Tschoegl, I. Emri: Generating line spectra from 3.11 M. Tao: High Temperature Deformation of Vitreloy experimental responses. Part III. Interconversion Bulk Metallic Glasses and Their Composite. Ph.D. between relaxation and retardation behavior, Int. Thesis (California Institute of Technology, Pasadena J. Polym. Mater. 18, 117–127 (1992) 2006) 3.26 I. Emri, N.W. Tschoegl: Generating line spectra from 3.12 J. Lu, G. Ravichandran, W.L. Johnson: Deforma- experimental responses. Part V. Time-dependent tion behavior of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk viscosity, Rheol. Acta. 36, 303–306 (1997) metallic glass over a wide range of strain-rates and 3.27 I. Emri, B.S. von Bernstorff, R. Cvelbar, A. Nikonov: temperatures, Acta Mater. 51, 3429–3443 (2003) Re-examination of the approximate methods for 3.13 W.G. Knauss: Viscoelasticity and the time- interconversion between frequency- and time- dependent fracture of polymers. In: Comprehen- dependent material functions, J. Non-Newton. Fluid sive Structural Integrity,Vol.2,ed.byI.Milne, Mech. 129, 75–84 (2005) R.O. Ritchie, B. Karihaloo (Elsevier, Amsterdam 2003) 3.28 A. Nikonov, A.R. Davies, I. Emri: The determina- 3.14 W. Flügge: Viscoelasticity (Springer, Berlin 1975) tion of creep and relaxation functions from a single 3.15 H. Lu, X. Zhang, W.G. Knauss: Uniaxial, shear and experiment, J. Rheol. 49, 1193–1211 (2005) Poisson relaxation and their conversion to bulk re- 3.29 M.A. Branch, T.F. Coleman, Y. Li: A subspace inte- laxation, Polym. Eng. Sci. 37, 1053–1064 (1997) rior, and conjugate gradient method for large-scale 3.16 N.W. Tschoegl, W.G. Knauss, I. Emri: Poisson’s ra- bound-constrained minimization problems, Siam J. tio in linear viscoelasticity, a critical review, Mech. Sci. Comput. 21, 1–23 (1999) Time-Depend. Mater. 6, 3–51 (2002) 3.30 F. Kohlrausch: Experimental-Untersuchungen über 3.17 I.L. Hopkins, R.W. Hamming: On creep and relax- die elastische Nachwirkung bei der Torsion, Aus- ation, J. Appl. Phys. 28, 906–909 (1957) dehnung und Biegung, Pogg. Ann. Phys. 8, 337 (1876) Mechanics of Polymers: Viscoelasticity References 93 3.31 G. Williams, D.C. Watts: Non-symmetrical dielec- 3.48 J. Bischoff, E. Catsiff, A.V. Tobolsky: Elastoviscous tric behavior arising from a simple empirical decay properties of amorphous polymers in the transition function, Trans. Faraday Soc. 66, 80–85 (1970) region 1, J. Am. Chem. Soc. 74, 3378–3381 (1952) 3.32 S. Lee, W.G. Knauss: A note on the determination of 3.49 F. Schwarzl, A.J. Staverman: Time-temperature de- relaxation and creep data from ramp tests, Mech. pendence of linearly viscoelastic behavior, J. Appl. Time-Depend. Mater. 4, 1–7 (2000) Phys. 23,838–843(1952) 3.33 A. Flory, G.B. McKenna: Finite step rate corrections 3.50 M.L. Williams, R.F. Landel, J.D. Ferry: Mechan- in stress relaxation experiments: A comparison of ical properties of substances of high molecular two methods, Mech. Time-Depend. Mater. 8,17–37 weight. 19. The temperature dependence of re- (2004) laxation mechanisms in amorphous polymers and 3.34 J.F. Tormey, S.C. Britton: Effect of cyclic loading on other glass-forming liquids, J. Am. Chem. Soc. 77, solid propellant grain structures, AIAA J. 1, 1763–1770 3701–3707 (1955) (1963) 3.51 D.J. Plazek: Temperature dependence of the vis- 3.35 R.A. Schapery: Thermomechanical behavior of vis- coelastic behavior of polystyrene, J. Phys. Chem. US coelastic media with variable properties subjected 69,3480–3487(1965) A Part to cyclic loading, J. Appl. Mech. 32, 611–619 (1965) 3.52 D.J. Plazek: The temperature dependence of the vis- 3.36 L.R.G. Treloar: Physics of high-molecular materials, coelastic behavior of poly(vinyl acetate), Polym. J. 12, Nature 181, 1633–1634 (1958) 43–53 (1980) 3.37 M.-F. Vallat, D.J. Plazek, B. Bushan: Effects of ther- 3.53 R.A. Fava (Ed.): Methods of experimental physics 3 mal treatment of biaxially oriented poly(ethylene 16C. In: Viscoelastic and Steady-State Rheological terephthalate), J. Polym. Sci. Polym. Phys. 24, 1303– Response, ed. by D.J. Plazek (Academic, New York 1320 (1986) 1979) 3.38 L.E. Struik: Physical Aging in Amorphous Polymers 3.54 L.W. Morland, E.H. Lee: Stress analysis for linear and Other Materials (Elsevier Scientific, Amsterdam viscoelastic materials with temperature variation, 1978) Trans. Soc. Rheol. 4, 233–263 (1960) 3.39 G.B. McKenna, A.J. Kovacs: Physical aging of 3.55 L.J. Heymans: An Engineering Analysis of Polymer poly(methyl methacrylate) in the nonlinear range Film Adhesion to Rigid Substrates. Ph.D. Thesis (Cal- – torque and normal force measurements, Polym. ifornia Institute of Technology, Pasadena 1983) Eng. Sci. 24, 1138–1141 (1984) 3.56 G.U. Losi, W.G. Knauss: Free volume theory and non- 3.40 A. Lee, G.B. McKenna: Effect of crosslink density on linear viscoelasticity, Polym. Eng. Sci. 32, 542–557 physical aging of epoxy networks, Polymer 29, 1812– (1992) 1817 (1988) 3.57 I. Emri, T. Prodan: A Measuring system for bulk 3.41 C. G’Sell, G.B. McKenna: Influence of physical and shear characterization of polymers, Exp. Mech. aging on the yield response of model dgeba 46(6), 429–439 (2006) poly(propylene oxide) epoxy glasses, Polymer 33, 3.58 N.W. Tschoegl, W.G. Knauss, I. Emri: The effect of 2103–2113 (1992) temperature and pressure on the mechanical prop- 3.42 M.L. Cerrada, G.B. McKenna: Isothermal, isochronal erties of thermo- and/or piezorheologically simple and isostructural results, Macromolecules 33,3065– polymeric materials in thermodynamic equilibrium 3076 (2000) – a critical review, Mech. Time-Depend. Mater. 6, 3.43 L.C. Brinson, T.S. Gates: Effects of physical aging 53–99 (2002) on long-term creep of polymers and polymer ma- 3.59 R.W. Fillers, N.W. Tschoegl: The effect of pressure on trix composites, Int. J. Solids Struct. 32,827–846 the mechanical properties of polymers, Trans. Soc. (1995) Rheol. 21, 51–100 (1977) 3.44 L.M. Nicholson, K.S. Whitley, T.S. Gates: The com- 3.60 W.K. Moonan, N.W. Tschoegl: The effect of pres- bined influence of molecular weight and tempera- sure on the mechanical properties of polymers. ture on the physical aging and creep compliance 2. Expansively and compressibility measurements, of a glassy thermoplastic polyimide, Mech. Time- Macromolecules 16, 55–59 (1983) Depend. Mat. 5, 199–227 (2001) 3.61 W.K. Moonan, N.W. Tschoegl: The effect of pres- 3.45 J.D. Ferry: Viscoelastic Properties of Polymers,3rd sure on the mechanical properties of polymers. 3. edn. (Wiley, New York 1980) Substitution of the glassy parameters for those of 3.46 J.R. Mcloughlin, A.V. Tobolsky: The viscoelastic be- the occupied volume, Int. Polym. Mater. 10, 199–211 havior of polymethyl methacrylate, J. Colloid Sci. 7, (1984) 555–568 (1952) 3.62 W.K. Moonan, N.W. Tschoegl: The effect of pressure 3.47 H. Leaderman, R.G. Smith, R.W. Jones: Rheology of on the mechanical properties of polymers. 4. Meas- polyisobutylene. 2. Low molecular weight polymers, urements in torsion, J. Polym. Sci. Polym. Phys. 23, J. Polym. Sci. 14,47–80(1954) 623–651 (1985) 94 Part A Solid Mechanics Topics 3.63 W.G. Knauss, V.H. Kenner: On the hygrothermo- by P.J. Blau, B.R. Lawn (American Society for Testing mechanical characterization of polyvinyl acetate, J. and Materials, Philadelphia 1986) pp. 90–108 Appl. Phys. 51, 5131–5136 (1980) 3.80 W.C. Oliver, G.M. Pharr: An improved technique for 3.64 I.Emri,V.Pavˇsek: On the influence of moisture determining hardness and elastic modulus using on the mechanical properties of polymers, Mater. load and displacement sensing indentation experi- Forum 16, 123–131 (1992) ments, J. Mater. Res. 7, 1564–1583 (1992) 3.65 D.J. Plazek: Magnetic bearing torsional creep ap- 3.81 I.N. Sneddon: The relation between load and pene- paratus, J. Polym. Sci. Polym. Chem. 6,621–633 tration in the axisymmetric boussinesq problem for (1968) a punch of arbitrary profile, Int. J. Eng. Sci. 3,47–57 3.66 D.J. Plazek, M.N. Vrancken, J.W. Berge: A torsion (1965) pendulum for dynamic and creep measurements 3.82 M. Oyen-Tiesma, Y.A. Toivola, R.F. Cook: Load- of soft viscoelastic materials, Trans. Soc. Rheol. 2, displacement behavior during sharp indentation of 39–51 (1958) viscous-elastic-plastic materials. In: Fundamentals 3.67 G.C. Berry, J.O. Park, D.W. Meitz, M.H. Birnboim, of Nanoindentation and Nanotribology II,ed.by atA Part D.J. Plazek: A rotational rheometer for rheological S.P. Baker, R.F. Cook, S.G. Corcoran, N.R. Moody studies with prescribed strain or stress history, J. (Mater. Res. Soc., Warrendale 2000) pp. Q1.5.1– Polym. Sci. Polym. Phys. Ed. 27, 273–296 (1989) Q1.5.6, MRS Proc. Vol. 649, MRS Fall Meeting, Boston 3.68 R.S. Duran, G.B. McKenna: A torsional dilatome- 3.83 L. Cheng, X. Xia, W. Yu, L.E. Scriven, W.W. Gerberich: 3 ter for volume change measurements on deformed Flat-punch indentation of viscoelastic material, J. glasses: Instrument description and measurements Polym. Sci. Polym. Phys. 38, 10–22 (2000) on equilibrated glasses, J. Rheol. 34, 813–839 (1990) 3.84 H. Lu, B. Wang, J. Ma, G. Huang, H. Viswanathan: 3.69 J.E. McKinney, S. Edelman, R.S. Marvin: Apparatus Measurement of creep compliance of solid polymers for the direct determination of the dynamic bulk by nanoindentation, Mech. Time-Depend. Mater. 7, modulus, J. Appl. Phys. 27,425–430(1956) 189–207 (2003) 3.70 J.E. McKinney, H.V. Belcher: Dynamic compressibility 3.85 G. Huang, B. Wang, H. Lu: Measurements of vis- of poly(vinyl acetate) and its relation to free volume, coelastic functions in frequency-domain by nanoin- J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 43–53 (1963) dentation, Mech. Time-Depend. Mater. 8,345–364 3.71 T.H. Deng, W.G. Knauss: The temperature and fre- (2004) quency dependence of the bulk compliance of 3.86 T.C.T. Ting: The contact stresses between a rigid in- Poly(vinyl acetate). A re-examination, Mech. Time- denter and a viscoelastic half-space, J. Appl. Mech. Depend. Mater. 1, 33–49 (1997) 33, 845–854 (1966) 3.72 S. Sane, W.G. Knauss: The time-dependent bulk 3.87 E.H. Lee, J.R.M. Radok: The contact problem for vis- response of poly (methyl methacrylate), Mech. Time- coelastic bodies, J. Appl. Mech. 27, 438–444 (1960) Depend. Mater. 5, 293–324 (2001) 3.88 E.Riande,R.Diaz-Calleja,M.G.Prolongo,R.M.Mase- 3.73 Z. Ma, K. Ravi-Chandar: Confined compression– gosa, C. Salom: Polymer Viscoelasticity-Stress and a stable homogeneous deformation for multiaxial Strain in Practice (Dekker, New York 2000) constitutive characterization, Exp. Mech. 40,38–45 3.89 J.R.M. Radok: Visco-elastic stress analysis, Q. of (2000) Appl. Math. 15, 198–202 (1957) 3.74 K. Ravi-Chandar, Z. Ma: Inelastic deformation 3.90 S.C. Hunter: The Hertz problem for a rigid spheri- in polymers under multiaxial compression, Mech. cal indenter and a viscoelastic half-space, J. Mech. Time-Depend. Mater. 4, 333–357 (2000) Phys. Solids 8,219–234(1960) 3.75 D. Qvale, K. Ravi-Chandar: Viscoelastic characteri- 3.91 W.H. Yang: The contact problem for viscoelastic bod- zation of polymers under multiaxial compression, ies, J. Appl. Mech. 33, 395–401 (1960) Mech. Time-Depend. Mater. 8,193–214(2004) 3.92 S.A. Hutcheson, G.B. McKenna: Nanosphere embed- 3.76 S.J.Park,K.M.Liechti,S.Roy:Simplifiedbulkexper- ding into polymer surfaces: A viscoelastic contact iments and hygrothermal nonlinear viscoelasticty, mechanics analysis, Phys. Rev. Lett. 94, 076103.1– Mech. Time-Depend. Mater. 8, 303–344 (2004) 076103.4 (2005) 3.77 A. Kralj, T. Prodan, I. Emri: An apparatus for 3.93 J.H. Teichroeb, J.A. Forrest: Direct imaging of measuring the effect of pressure on the time- nanoparticle embedding to probe viscoelasticity of dependent properties of polymers, J. Rheol. 45, polymer surfaces, Phys. Rev. Lett. 91, 016104.1– 929–943 (2001) 106104.4 (2003) 3.78 J.B. Pethica, R. Hutchings, W.C. Oliver: Hardness 3.94 M.L. Oyen: Spherical indentation creep following measurement at penetration depths as small as ramp loading, J. Mater. Res. 20, 2094–2100 (2005) 20 nm, Philos. Mag. 48, 593–606 (1983) 3.95 M.R. VanLandingham, N.K. Chang, P.L. Drzal, 3.79 W.C. Oliver, R. Hutchings, J.B. Pethica: Measure- C.C. White, S.-H. Chang: Viscoelastic characteriza- ment of hardness at indentation depths as low as tion of polymers using instrumented indentation–1. 20 nanometers. In: Microindentation Techniques in Quasi-static testing, J. Polym. Sci. Polym. Phys. 43, Materials Science and Engineering ASTM STP 889,ed. 1794–1811 (2005) Mechanics of Polymers: Viscoelasticity References 95 3.96 J.L. Loubet, B.N. Lucas, W.C. Oliver: Some meas- 3.111 R.A. Schapery: A theory of non-linear thermovis- urements of viscoelastic properties with the help coelasticity based on irreversible thermodynamics, of nanoindentation, International workshop on In- Proc.5thNatl.Cong.Appl.Mech.(1966)pp.511–530 strumental Indentation, ed. by D.T. Smith (NIST 3.112 R.A. Schapery: On the characterization of nonlinear Special Publication, San Diego 1995) pp. 31–34 viscoelastic materials, Polym. Eng. Sci. 9,295–310 3.97 B.N. Lucas, W.C. Oliver, J.E. Swindeman: The dy- (1969) namic of frequency-specific, depth-sensing inden- 3.113 H.Lu,W.G.Knauss:Theroleofdilatationinthe tation testing, Fundamentals of Nanoindentation nonlinearly viscoelastic behavior of pmma under and Nanotribology, Vol. 522, ed. by N.R. Moody multiaxial stress states, Mech. Time-Depend. Mater. (Mater. Res. Soc., Warrendale 1998) pp. 3–14, MRS 2, 307–334 (1999) Meeting, San Francisco 1998 3.114 H. Lu, G. Vendroux, W.G. Knauss: Surface defor- 3.98 R.J. Arenz, M.L. Williams (Eds.): A photoelastic tech- mation measurements of cylindrical specimens by nique for ground shock investigation. In: Ballistic digital image correlation, Exp. Mech. 37, 433–439 and Space Technology (Academic, New York 1960) (1997) 3.99 Y. Miyano, T. Tamura, T. Kunio: The mechanical and 3.115 W.H. Peters, W.F. Ranson: Digital imaging tech- A Part optical characterization of Polyurethane with appli- niques in experimental stress analysis, Opt. Eng. 21, cation to photoviscoelastic analysis, Bull. JSME 12, 427–432 (1982) 26–31 (1969) 3.116 M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, 3.100 A.B.J. Clark, R.J. Sanford: A comparison of static and S.R. McNeil: Determination of displacements using 3 dynamic properties of photoelastic materialsm, Exp. an improved digital image correlation method, Im. Mech. 3,148–151(1963) Vis. Comput. 1, 133–139 (1983) 3.101 O.A. Hasan, M.C. Boyce: A constitutive model for 3.117 H. Lu: Nonlinear Thermo-Mechanical Behavior of the nonlinear viscoelastic-viscoplastic behavior of Polymers under Multiaxial Loading. Ph.D. Thesis glassy polymers, Polym. Eng. Sci. 35, 331–334 (1995) (California Institute of Technology, Pasadena 1997) 3.102 J.S. Bergström, M.C. Boyce: Constitutive modeling of 3.118 D.R. Lide: CRC Handbook of Chemistry and Physics the large strain time-dependent behavior of elas- (CRC Press, Boca Raton 1995) tomers, J. Mech. Phys. Solids 46, 931–954 (1998) 3.119 G.B. McKenna: On the physics required for predic- 3.103 P.D. Ewing, S. Turner, J.G. Williams: Combined tion of long term performance of polymers and their tension-torsion studies on polymers: apparatus and composites, J. Res. NIST 99(2), 169–189 (1994) preliminary results for Polyethylene, J. Strain Anal. 3.120 P.A. O’Connell, G.B. McKenna: Large deforma- Eng. 7,9–22(1972) tion response of polycarbonate: Time-temperature, 3.104 C. Bauwens-Crowet: The compression yield behavior time-aging time, and time-strain superposition, of polymethal methacrylate over a wide range of Polym. Eng. Sci. 37, 1485–1495 (1997) temperatures and strain rates, J. Mater. Sci. 8,968– 3.121 J.L. Sullivan, E.J. Blais, D. Houston: Physical ag- 979 (1973) ing in the creep-behavior of thermosetting and 3.105 L.C. Caraprllucci, A.F. Yee: The biaxial deformation thermoplastic composites, Compos. Sci. Technol. 47, and yield behavior of bisphenol-A polycarbonate: 389–403 (1993) Effect of anisotropy, Polym. Eng. Sci. 26,920–930 3.122 I.M. Hodge: Physical aging in polymer glasses, Sci- (1986) ence 267, 1945–1947 (1995) 3.106 W.G. Knauss, I. Emri: Non-linear viscoelasticity 3.123 N. Goldenberg, M. Arcan, E. Nicolau: On the most based on free volume consideration, Comput. Struct. suitable specimen shape for testing sheer strength 13, 123–128 (1981) of plastics, ASTM STP 247, 115–121 (1958) 3.107 W.G. Knauss, I. Emri: Volume change and the non- 3.124 M. Arcan, Z. Hashin, A. Voloshin: A method to pro- linearly thermo-viscoelastic constitution of poly- duce uniform plane-stress states with applications mers, Polym. Eng. Sci. 27, 86–100 (1987) to fiber-reinforced materials, Exp. Mech. 18, 141–146 3.108 G.U. Losi, W.G. Knauss: Free volume theory and (1978) nonlinear thermoviscoelasticity, Polym. Eng. Sci. 32, 3.125 M. Arcan: Discussion of the iosipescu shear test as 542–557 (1992) applied to composite materials, Exp. Mech. 24, 66– 3.109 G.U. Losi, W.G. Knauss: Thermal stresses in nonlin- 67 (1984) early viscoelastic solids, J. Appl. Mech. 59, S43–S49 3.126 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic be- (1992) havior of polycarbonate. I. Response under pure 3.110 W.G. Knauss, S. Sundaram: Pressure-sensitive dis- shear, Mech. Time-Depend. Mater. 6, 231–269 (2002) sipation in elastomers and its implications for the 3.127 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic be- detonation of plastic explosives, J. Appl. Phys. 96, havior of polycarbonate. II. The role of volumetric 7254–7266 (2004) strain, Mech. Time-Depend. Mater. 6, 301–322 (2002)