Thermal activation based constitutive equations for polymers

Size: px

Start display at page:

Download "Thermal activation based constitutive equations for polymers"

Maurice Goodman
6 years ago
Views:

1 J. Phys. IVFrance 10 (2000) O EDP Sciences, Les Ulis Thermal activation based constitutive equations for polymers F.J. Zerilli and R.W. Armstrong* Naval Surface Warfare Center Indian Head Division. lndian Head. MD U.S.A. * Depattment of Mechanical Engineering, ~niversify of Maryland, 'college Park, MD 20742, U.S.A. Abstract. Minimal assumptions are used in deriving constitutive equations for polymers based on the theory of thermal activation. With a shear volume of activation inversely proportional to the flow stress, the general structure of the constitutive equation as a function of temperature, pressure, and strain rate may be elucidated without regard to the details of the underlying flow mechanisms. Strain hardening or softening is described by a differential equation relating the hardening rate to creation or destruction of flow units. While many polymers are, strictly speaking, nonlinearly viscoelastic materials, over a limited time and temperature range they may be considered to exhibit unrecoverable, though not necessarily volume conserving, strains and treated as viscoplastic. The equations have been applied to the yield stress of glassy PMMA, showing the excellent result that may be obtained with the assumed inverse dependence of the shear volume of activation upon the flow stress, and to the dynamic deformation behavior of semicrystalline PTFE, showing a reasonably good reproduction of the stress-strain behavior as a function of temperature, pressure, and strain rate. The yield stress results for PMMA are used to describe its ductile-brittle transition properties. 1. INTRODUCTION We have developed a constitutive equation for polymers which addresses the temperature, strain-rate, and pressure dependence of the stress-strain relation. The small plastic component of strain in hydrostatic compression is not addressed, nor are such factors as brittleness and fracturing associated with strain-rate, temperature, and pressure effects. It is assumed that the strain-rates and temperatures are in the range in which creep and long-term relaxation effects are negligible. The model is isotropic and one dimensional and may be considered to describe the relation between Mises effective stress and strain. However, by including pressure dependence the model is able to account for the observed differences between compressive and tensile loading. The extension to three dimensional non-isotropic conditions is left for future work. A more complete exposition will be presented elsewhere [I]. The total strain is divided into a viscoplastic part and a viscoelastic part, the viscoplastic part being described by a non-linear thermal activation dashpot, and the viscoelastic part being described in terms of a Maxwell-Weichert linear viscoelasticity. 2. CONSTITUTIVE EQUATION 2.1. Thermal Activation Model Following the pioneering work of Eyring[2], an Arrhenius form of thermally activated constitutive equation for the temperature and strain rate dependence of flow stress has been well established for a wide variety of solid materials over appropriate temperature and strain rate regimes. The thermal activation model describes Article published online by EDP Sciences and available at

2 Pr9-4 JOURNAL DE PHYSIQUE IV the rate of any process in which matter rearranges by surmounting a potential energy barrier. When the potential barrier is suff~ciently high, the rate in the forward direction exceeds the reverse rate significantly and the plastic strain rate is given by the expression[3] -G/U E = eoe (1) where G is the activation energy for the reaction, kis Boltzmann's constant, and T is the absolute temperature. G may be identified with the Gibbs free energy, so we may write the standard thermodynamic relation dg = -SdT - yjdoo (2) where S is the entropy and the volumes of activation Kj are the thermodynamic variables conjugate to the stresses oij At constant temperature, specializing to isotropic materials, and assuming that there is no dependence on the third stress invariant, Eq. (1) may now be written JVado = [v& + Go + k~ln(&l&,,). (3) where we assume that Va is a function of effective stress o and pressurep and that V, is a function only of pressure. Assuming that the effective stress activation volume depends inversely on the effective stress, and solving for the effective stress, Eq. (3) becomes = B(p)e - P@& where the function B(p) is the flow stress at T = 0 K and 2.2. Viscoplastic Stress-Strain Behavior of Polymers In order to describe the viscoplastic stress-strain behavior of polymers, two thermally activated processes are considered, one associated with the initial yield behavior, the other associated with subsequent strain hardening. Thus the total flow stress is written o = B(p)e-P@,E)T + 8(eg)e - a@, &)T (7) where two terms of the form of Eq. (5) have been included. The presence of two separate deformation mechanisms was suggested in a differential scanning calorimetry study of glassy polymers by Hasan and Boyce[4]. Two distinct exotherms were found, one, at temperatures below the glass transition temperature, associated with the initial yield and strain softening behavior, the other, at temperatures above the glass transition temperature, associated with the strain hardening behavior Strain Hardening in Polymers A number of processes may play a role in the plastic or elastic deformation of polymers: chain slippage, kinking and un-kinking of chains, chain scission, cross-linking, chain entanglement, chain segment rotation. One or more of these processes will be important for a given polymer in a certain regime of temperature, strain-rate, and pressure. Whatever the processes, we describe the aggregate result abstractly in terms of units offlow as defrned by Kauzmann[S]: "structures in a body whose motions past one another make up the unit shear stress process... the unit of flow may be a single molecule or a group of many molecules, and the barrier may arise directly fiom the repulsions between a few molecules or tom some more complicated mechanism". Here we consider an areal density of units of flow, p, which are somewhat analogous to dislocations in crystalline materials -- line defects which are the boundaries between slipped and unslipped regions.

3 EURODYMAT 2000 Pr9-5 Gilman[6] has discussed the application of dislocation theory to amorphous materials and the need to replace the constant Burgers displacement in crystalline materials with the average of a fluctuating Burgers displacement in the amorphous material. Extending the analogy further, we assume that, at low temperatures, in the absence of thermal fluctuations, the flow stress is proportional to the square root of p e = a@ (8) where is a constant. Following Bergstrom's analysis for dislocation interactions in iron[7], we divide the flow unit density into a mobile flow unit density pm and an immobile flow unit density pi and assume that the mobile flow unit density is constant. Then the total flow unit density may be related to the plastic strain by the differential equation where b is the average displacement produced by a flow unit, h is the mean fiee path for immobilization of flow units, and o is the rate for mobilization of flow units. We allow o to be negative, this term then also contributing to the immobilization of flow units. If b, h, and o are constant, the equation is easily solved and the cold flow stress becomes 6 = B~&F&. (10) At temperatures above zero, this cold flow stress is reduced by the thermal activation factor e Total Viscoplastic Component of Deformation The resulting viscoplastic component of the deformation is described by the equation where a = ~ e - ~ ~ + (1 1) and, neglecting their potential pressure dependence, Po, PI, a,, a, are constants. From available experimental data for polytetrafluoroethylene it was determined that o depends approximately logarithmically on the strain rate and linearly on the pressure, so we choose the functional form o = o + oblne + o 2' (13) where a,, a,,o, are constants. The form of the pressure dependence for the coeff~cients B and B, follows the result of Argon[8] who derived the expression T = (a + (14) o for the low temperature shear yield stress of glassy polymers as a function of pressure. Argon's model is based on the thermally activated production of pairs of kinks in the collection of interpenetrating smooth chain molecules comprising the polymer. Thus, we choose? B = Bpa(l + ipbpfp' c~ (15) B, = Bop(' + B~~~~~~~ Fieure - 1. Maxwell-Weichert linear visco- where B,,, Bpb, Bpn, BOP, Bopb, BOP,, are constants. 1 elasticity plus thermal activation non-linear viscoplasticity.

4 Pr9-6 JOURNAL DE PHYSIQUE IV 2.5. Polymer Constitutive Equation: Viscoelastic Component A Maxwell-Weichert model is used to describe the initial viscoelastic stress. This model is placed in series with the non-linear dashpot described by Eq. (1 1) above as illustrated in Fig. 1. Thus the basic equations which describe the model are: where a (E &> is the hction in Eq. (11) relating stress to strain and strain rate, P P' Oi = E~E~) = qi&y where E,, q, are the modulus and viscosity of the i'th component, and = &(e) + E(V) + & i r P where E, E:), ep), E are the total strain, the i'th component elastic strain, the i'th component viscous strain, P and the viscoplastic strain, respectively. The viscosities, or equivalently, the relaxation times, zi = q,/e,, have a temperature and pressure dependence given by H,IT T, = zoie (19) where Hi = Hoi + A$. (20) 3. APPLICATIONS e-* 3.1. The Yield Stress of Polymethylmeth- & acryiate &- e-3 The thermal activation model has been applied to the yield stress of polymethylmethacrylate e-4 (PMMA) at various temperatures and strain I0000 rates[9] as measured by Bauwens-Crowet [lo] and it has been shown that a volume of activation (K) inversely proportional to the yield stress gives good agreement between the and measured yield stresses. In Fig. 2, is plotted Bauwens-Crowet's data for the compressive yield stress of PMMA divided by its shear modulus as a function ofthe e-' Figure 2. Compressive yield stress divided by shear modulus versus Oik = ~ln(iji) for PMMA. Data of Bauwens-Crowet (Ref. 10) is shown by the symbols. The solid line is the model fit for a volume of activation inversely proportional to effective shear stress, while the dotted curve is the result of a fit for a more general inverse power dependence. quantity ~ln(&,/&) = Glk. With the correct choice of E0, the equality holds, the quantity is the activation energy in degrees Kelvin, and the data in the plot should coalesce into a single one-parameter curve, as it does in Fig. 2 with the choice of E0= 2 x lo7 s-i. Furthermore, if the volume of activation is inversely proportional to the effective shear stress, then the curve in an ln(stress) plot should be a straight line with slope -klwo and intercept In(B1p). An inspection of Fig. 2 shows this to be very closely the case, with WWo = 2.56 x K-' andbip = 0,454. These parameters then give calculated values [9] of yield stress versus strain rate and temperature which match well with Bauwens-Crowet's data The PMMA Ductile-Brittle Transition The yield stress results obtained in compression for PMMA at different temperatures and strain rates may be

$EURODYMAT 2000 Pr9-7 compared with predictions of a transition to brittle fracturing when tested in tension.$

5 EURODYMAT 2000 Pr9-7 compared with predictions of a transition to brittle fracturing when tested in tension. The pre-cracked brittle fracture stress of PMMA, oc, is known to follow the relation oc = 4[s/(~+s)]112, (21) where ot is the crack-free tensile fracture stress and s is the width of an assumed plastic-type strip zone at the tip of a crack of radius c. The PMMA 120 fracture stress data cover a range of crack sizes such that s was determined by the reciprocal square root of c dependence at large c values and ' 80 approached asymptotically the value of o, at $ small crack sizes [ A tensile ductile-brittle transition may be defined for PMMA and related polymers by the 240 condition 20 o$t,e) = oc(t,e), (22) 0 - where at larger crack sizes oc is normally taken on a fracture surface energy basis to have a TRUE TOTAL STRAIN temperature and strain rate Figure 3. Total stress-strain calculated with Eqs. (11) and dependence. At smaller crack sizes, a somewhat (16-20), compared with split Hopkinson pressure bar data larger temperature and strain rate dependence reported by Walley and Field for PTFE (Ref. 12). may enter for oc. Equation (22) provides a means of calculating the ductile-brittle transition temperature as a function of strain rate. Thus, for example, for an intentional crack radius of 1.0 mm., and strain rate of s-', brittle tensile fracture should occur before yield at all temperatures less than 3 13 K, whereas for a smaller unintentional crack radius of 0.04 mm, and strain rate of 1.0 s-', brittle fracture would occur at yield for T K, and before yield, at temperatures less than 253 K Constitutive Equations for Polytetrafluoroethylene The parameters for use in the constitutive equation for polytetrafluoroethylene (PTFE) were obtained by analyzing results reported by anumber of investigators: for the viscoplastic part, compressive split Hopkinson pressure bar stress-strain curves at a number of strain-rates reported by Walley and Field[l2], a low temperature (130 K) Hopkinson bar stress- A strain curve reported by Walley, Field, Pope, and 100 / ' / PTFE Safford[l3], Hopkinson bar data at a number of a 294 K temperatures and strain-rates determined by s ' Gray [l4], and tensile stress-strain data at various 0 I MPa superposed hydrostatic pressures reported by r 138MPa - Sauer and Pae[l5]. An eight component Maxwell-Weichert model was used for the viscoelastic part, with the moduli, relaxation.*..***** times, and activation energies, El, zo,, HOI, respectively, chosen to give a reasonably good match to the shear modulus and logarithmic decrement vs. temperature data at 1 Hz published by McCrum[l6]. The complete analysis is presented elsewhere [I]. ~i~~~~ 3 and 4 show stress-strain curves calculated with Eqs. (11) and (16-20) compared 276 MPa 552 MPa TRUE TOTAL STRAIN Figure 4. Total stress-strain calculated with Eqs. (1 I) and (16-20). for different pressures, compared with tensile test data reported by Sauer and Pae (Ref. IS).

Pr9-8 JOURNAL DE PHYSIQUE IV to the reported experimental data. In Fig. 3, the compressive Hopkinson bar data for stress vs. strain at six strain rates ranging from 0.

6 Pr9-8 JOURNAL DE PHYSIQUE IV to the reported experimental data. In Fig. 3, the compressive Hopkinson bar data for stress vs. strain at six strain rates ranging from s-' to s-' reported by Walley and Field are compared to the calculated results. The change in shape of the curves with strain rate is well reproduced. The increased strain hardening with strain corresponds to negative values of o in Eq. (1 I), o becoming more negative with increasing strain rate. In Fig.4, tensile stress-strain with superimposed hydrostatic pressure data reported by Sauer and Pae are compared to calculated results. The change of shape of the curves with pressure is well reproduced. 4. CONCLUSION The thermal activation model with a volume of activation inversely proportional to the effective shear stress gives an excellent description of the yield stress of PMMA as a function of temperature and strain rate. To achieve good results, it is important to take into account the dependence of the shear modulus on temperature, and probably strain rate as well. The thermal activation model also provides a basis for describing the yield stress and strain hardening in PTFE, with additional strain hardening accounted for by a pressure and strainrate dependent rate of destruction of flow units. The constitutive model describes not only the temperature and strain rate dependence of the flow stress, but also its pressure dependence and, therefore, should be able to account for the observed differences in polymers between compressive and tensile loading. Acknowledgments This work was supported at NSWC by the In-house Laboratory Independent Research Program and at the University of Maryland by NSWC and the Office of Naval Research. References [I] Zerilli, F. J., and Armstrong, R. W., article in preparation. [2] Eyring, H., J: Chem. Phys. 3 (1935) 107; Eyring, H., J. Chem. Phys. 4 (1936) 283. [3] Escaig, B., Ann. Phys. 3 (1978) [4] Hasan, 0. A., and Boyce, M. C., PoZymer 34 (1993) [5] Kauzmann, W., Trans. Am. Znst. Min. Metall. Eng. 143 (1941) 57. [6] Gilman, J. J., J: Appl. Phys. 44 (1973) 675. [7] Bergstrom, Y., Mat. Sci. Eng. 5 (1970) [8] Argon, A. S., Philos. Mag. 28 (1973) 839. [9] Zerilli, F. J., and Armstrong, R. W., Proceedings of the American Physical Society 1999 Topical Conference on Shock Compression of Condensed Matter, Snowbird, Utah, 27 June- 2 July 1999, to be published. [lo] Bauwens-Crowet, C., J: Mater. Sci. 8 (1973) 968; Fotheringham, D. and Cherry, B. W., Discussion, J. Mater. Sci. 11 (1976) 1368, and Reply, Ibid., p [l 11 Armstrong, R. W., in "Third International Conference on Fracture", A. Kochendorfer, Ed. (Verein Deutscher Eisenhuttenleute, Munich, 1973), Paper [12] WaHey, S. M., and Field, J. E., DYMAT Journal 1 (1994) , Fig. 2 o Walley, S. M., Field, J. E., Pope, P. H., and Safford, N. A., J. Phys. ZIZFrance 1 (1991) 1889, Fig [14] Gray, G. T., 111, Cady, C. M., and Blumenthal, W. R., in Constitutive and Damage Modeling of Inelastic Deformation and Phase Transformation, Proceedings of Plasticity '99, The Seventh International Symposium on Plasticity and Its Current Applications, Akhtar S. Khan, Ed. (Neat Press, Fulton, Maryland, 1999) p [IS] Sauer, J. A., and Pae, K. D., Colloid & Polymer Sci. 252 (1974) 680. [16] McCrurn, N. G., J: Polymer Sci. 34 (1 959) 355.

What we should know about mechanics of materials

What we should know about mechanics of materials 0 John Maloney Van Vliet Group / Laboratory for Material Chemomechanics Department of Materials Science and Engineering Massachusetts Institute of Technology