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Nonlinear Viscoelastic and Viscoplastic Response of Glassy Polymers.

The main features of inelastic mechanical behavior of glassy state were studied theoretically and experimentally in terms of tensile stress-strain and tensile creep experiments. A theoretical treatment introduced in earlier work, which takes into account the viscoelastic path at small strains and the viscoplastic one at higher stresses, proved to be capable of describing the main aspects of mechanical response of glassy polymers, i.e. nonlinear viscoelasticity during creep procedure, and yield stress, yield strain, strain softening and rate effect in a constant crosshead speed test.


Solid polymers are widely used in load carrying applications, both in the homogeneous state and as a matrix phase in polymeric composite materials. Therefore, the prediction of their inelastic mechanical behavior, in terms of monotonic and cycling loading, as well as creep and relaxation, is of great importance. Numerous investigations have been focused on the nonlinear viscoelasticity [1-6] and viscoplastic response of polymers [7-11]. This response is strongly rate and temperature dependent and thermomechanicaly prehistory of the material dependent. An interesting paper dealing with the long term material performance in short-term stress relaxation tests on polycarbonate has been published recently [12].

However, until now, all these aspects of deformation behavior have been treated separately. A number of interesting approaches deal with nonlinear viscoelastic behavior, using integral representation with multiple relaxation times, stress dependent, or using state variables related to the free volume [1]. These treatments do not consider the yield and post-yield behavior of solid polymers, which are thoroughly studied in other works. Therefore, as is also mentioned by Krempl [13], it is necessary to develop a unified model that represents the nonlinear viscoelastic, yield and post-yield viscoplastic response of solid polymers.

A recent work by Hasan and Boyce [14] presents a model that can describe both experimental results of strain rate compression tests at different temperatures, and creep tests at various stress levels and temperatures. In their approach they use a constitutive model based on the distributed nature of the microstructural state and the thermally activated evolution of the glassy state. The rate of plastic deformation is considered to be the result of a thermal activated rearrangement, based on the Arrhenius formulation, modified by argon. However, through their analysis two main difficulties arise, acknowledged by the authors themselves: the inadequacy of predicting both the exact position of yield strain and the strain softening. Therefore, in order to describe in detail yield and post-yield behavior, the authors are forced to Introduce many phenomenological equations, which describe the time evolution of several internal parameters in a complicated manner.

In this work, a constitutive model, developed previously [15], leads to the evaluation of the rate of plastic deformation in glassy polymers, based on the distribution of plastic shear transformations in a completely different way: What is distributed is the strain, and, in respect to the overall state of strain, the fraction of sites that have been plastically transformed can be calculated. It is also important to mention that the maximum rate of plastic deformation takes place at the yield strain.

A shortcoming of this approach is the fact that the rate of plastic deformation is stress independent, and therefore the rate effect cannot be predicted. To overcome this difficulty, we take into account that in a constant crosshead-speed experiment, at small strains the viscoelastic path is dominant; therefore, the procedure can be treated as a thermally activated one, and applying the corresponding constitutive equations of viscoelasticity , the rate effect can be easily described. When the deformation reaches a critical value (which acts as a control parameter) the plastic path prevails. At this stage, the decomposition of strain into viscoelastic and plastic part will be made in terms of a kinematic formulation introduced by Rubin (16), while a mechanism leading to the quantitative evolution of the rate of plastic deformation, proposed elsewhere (15), will also be used.

Through this analysis, the exact position of yield strain, yield stress and strain softening can be predicted in a self-consistent manner. Moreover, this model was proved to be capable of identifying the nonlinear viscoelastic response of the glassy state during creep experiments, as will be shown.


The material tested was polycarbonate, a product of GE Plastics with the commercial name Lexan, provided in plate form. In order to eliminate any prehistory effects, the samples were annealed at a temperature above [T.sub.g] for 4 hours. Creep experiments were conducted at room temperature, at various stress levels, while the deformation could be measured very accurately at every localized region, along the total gauge length. The experimental procedure followed, is based on a noncontact method with a laserextensometer described in detail elsewhere (15). Typical creep curves were then obtained as presented in Fig. 1. The instantaneous response is followed by inelastic flow at a decreasing rate (primary and secondary creep) when the imposed stress is quite lower than the yield stress. For high stress levels the inelastic strain rate increases rapidly, resulting in the failure (flow) of the material. The experimental data of Fig. 1 are replotted on a double logarithmic scale, and shown in Fig. 2, where the earl y time response has been omitted. As is known, the material response at this stage is considered to be unreliable, owing to rise time effects.

Apart from creep tests, tensile experiments have been carried out with an Instron tester 1121 at room temperature, at three different crosshead speeds, namely 0.1, 1 and 10 mm/min. which coincide with a total effective strain rate equal to 5.55 X [10.sup.-5], 5.55 x [10.sup.-4] and 5.55 x [10.sup.-3] [sec.sup.-1] respectively. The longitudinal strain was measured with the above mentioned experimental method. True stress strain curves were then constructed in respect to that part of the gauge length that exhibits the maximum deformation. These plots are presented in Fig. 3. More details about these experiments are reported eleswhere (15). The main characteristics of typical mechanical response are exhibited in these curves, such as initial viscoelastic behavior, followed by a nonlinear viscoelastic one, up to the stress overshoot and subsequent strain softening and strain hardening. The rate effect is also obvious, and this fact may be attributed to the thermally activated process during inelastic deformation of glassy polymers.


A quantitative description of creep curves has been made by Mindel and Brown (17), in which creep results of PMMA were represented by an equation of the general form:

[epsilon] = [f.sub.1](T) [f.sub.2]([sigma]/T) [f.sub.3]([epsilon]) (1)

According to the theory of thermal activation as proposed by Eyring (18), used by Bauwens-Crowet et al. (19), the creep rate takes the following form:

[epsilon] = constante [e.sup.-Q/KT] [e.sup.-[sigma][v.sub.p]/3KT] [e.sup.([sigma]-[sigma][])[v.sub.a]/KT (2)

where Q is the activation energy, [sigma] is the stress, [v.sub.p] [v.sub.s] are the pressure and shear activation volumes respectively. [[sigma]] is an internal stress that opposes the applied stress, and is proportional to the amount of strain that will be recovered:

[[[sigma]] = [k.sub.2] [[epsilon].sub.R] (3)

where [k.sub.2] is a constant proportional to the temperature and [[epsilon].sub.R] the recoverable strain. Because [[sigma]] increases with strain, [sigma]-[[sigma]] also decreases with strain, and therefore, following Eq 2, creep rate reaches a minimum value, as is also shown by the experimental results.

In this work polycarbonate has been tested in creep experiments for various stress levels at room temperature. In Fig. 1 creep strain versus time is presented for different stresses. As is shown, even in low stresses, the instantaneous part of creep strain is a nonlinear function of stress, while the other part of the creep curve remains parallel for the various stresses. The instantaneous strain is followed by the primary creep stage, where the creep rate is decreasing, followed by an almost stable creep rate, while at high stresses (close to the yield stress) an abrupt change of the slope takes place, related to creep failure.

The data of Fig. 1 are also plotted in respect to creep rate [epsilon] versus strain [epsilon], in Fig. 4. The initial part of this curve indicates the decreasing creep rate during the early stages of creep procedure, while the minimum of creep rate, or equivalently the onset of accelerated creep takes place at a certain value of strain equal to 0.055.

The shape of this curve remains quite the same for the various stress levels examined, and the effect of stress is exhibited by a slight shifting of strain value at which the minimum creep rate or the beginning of accelerated creep occurs.

Equations 2 and 3 were combined with the experimental data of Fig. 4 where creep rate [epsilon] versus [epsilon] is presented. Applying the nonlinear fit procedure of the software Mathematica (20), parameters [k.sub.1], [k.sub.2], [v.sub.p], [v.sub.s] were calculated, where [k.sub.1] = constant exp [-Q/kT]. It is important to mention that the various curves were fitted with the same set of parameters, indicating the validity of the model as well as the molecular and physical meaning of these quantities. Parameter values were found to be as follows: [k.sub.1] = [10.sup.-18.6] [(sec).sup.-1], [k.sub.2] = 770 MPa, [v.sub.p]/3KT = 0.27 [(Mpa).sup.-1], [v.sub.s]/kT = 0.34 [(Mpa).sup.-1].

In Fig. 5, for a representative stress level equal to 61 MPa, the corresponding experimental data were plotted up to the minimum of this curve, denoting that the plastic part of total strain is still almost negligible at this stage, and are shown in respect to the theoretical (fitted) curve.

In the second stage of this analysis, it will be shown that the above-mentioned model is capable of predicting the stress-strain behavior as well.

Making the assumption that stress-strain behavior of the glassy state follows a thermally activated viscoelastic path at small strains, a well known constitutive equation from viscoelasticity will be used:

[sigma](t) = [[[integral].sup.t].sub.0] [E.sub.0] exp([xi] - [xi]') de/d[lambda] d[lambda] (4)

where [E.sub.0] is the apparent modulus equal to 2300 Mpa, and [xi], [xi]' is the reduced time given by:

[xi] = [[[integral].sup.t].sub.0] dt'/[alpha]([sigma](t')) and [xi]' = [[[integral].sup.[lambda]].sub.0] dt'/[alpha]([sigma](t')) (5)

The shift factor [alpha] is the ratio [[tau].sub.r]/[tau] of the relaxation time in a reference state of deformation over the relaxation time [tau] in every stage of stress. The relaxation time [tau] has been estimated according to the Eyring treatment (18) from the condition [epsilon][tau] [congruent] 1. Therefore, following Eq 2, the relaxation time is given by the expression:

[tau] = [[K.sup.-1].sub.1] exp[-[sigma][v.sub.p]/3kT] exp[-([sigma] - [[sigma]][v.sub.s]/kT] (6)

In order to describe the yield and post-yield behavior after the initial viscoelastic path, we assume that at this stage plastic path prevails. Therefore there is a need to consider the plastic strain that starts developing during this stage. A kinematic formulation that separates the total deformation into plastic and elastic parts, developed by Rubin (16), will be applied. Rubin specified an evolution equation for the elastic deformation, including the relaxation effect of plastic deformation, without introducing a plastic deformation tensor explicitly. For uniaxial deformation the following expression for the time evolution of the stretch ratio [a.sub.m] = 1 + [epsilon] of the viscoelastic deformation has been obtained (21):

[a.sub.m]/[a.sub.m] = f([a.sub.m]) [a/a - [[gamma].sub.p]/18 g)[a.sub.m])] (7)

where f([a.sub.m]) and g([a.sub.m]) are specific functions of the elastic strain [a.sub.m], a is the imposed strain rate, a is the stretch ratio, v is the Poisson ratio and [a.sub.m](O) = 1. The quantity [[gamma].sub.p] expresses the rate of plastic deformation and will be specified in the following.

Moreover, a mechanism of rate of plastic deformation introduced by Oleinik (22, 23) and developed in quantitative terms (15) will be taken into account. Following this mechanism, the imposition of a stress field into the material results in a distribution of strain to selective regions characterized by high amounts of free volume. At these specific regions, formation and growth of plastic shear transformations (PSTs) can further take place. Owing to the inhomogeneous strain field around each PST, they are of varying size, following a continuous distribution described by a certain probability density in respect to the size of PSTs as a random variable.

Extending this concept, it is further assumed that an analogous distribution density will be followed by the strains evolved around PSTs, through which the appropriate amount of elastic energy is offered for the PSTs' transition to a nonrecoverable state. It is reasonable to suppose that the size of PSTs follows a normal distribution, and equivalently it may be assumed that strain field [[epsilon].sub.1] around them follows a normal distribution as well.

Hence, the distribution density (normal distribution) of the PSTs in respect to strain [[epsilon].sub.1] as a random variable is given by:

F([[epsilon].sub.1]) = 1/s[square root of]2[phi] [e.sup.-1/2] [([[epsilon].sub.t]-[micro]/s).sup.2] (8)

where [micro] is the mean value and s is the standard deviation of the strain field. The application of strain field with an [epsilon] rate activates the process of nucleation, growth and merging of PST, leading in some cases to an irreversible state, that is, to plastic deformation. The fraction of PST that has enough activation energy to attain a new nonreversible state is given by the probability:

P(0 [less than] [[epsilon].sub.t] [less than or equal to] [epsilon]) = F([epsilon]) - F(0) (9)

where F([epsilon]) expresses the distribution function. Therefore, Eq 9 becomes:

P([epsilon]) = F([epsilon]) - F(0) = 1/s[square root]2[pi] [[[integral].sup.[epsilon]].sub.0] [e.sup.-1/2[([[epsilon].sub.i]-[micro]/s).sup.2]] d[[epsilon].sub.i] (10)

Making the further assumption that the rate of plastic deformation [[gamma].sub.p] is proportional to the fraction of PSTs that have achieved a nonreversible state, and this transition takes place with an average rate k for every PST, we have:

[[gamma].sub.p] = k P(0 [less than] [[epsilon].sub.i] [less than or equal to] [epsilon]) (11)

The value of k can be estimated, assuming that at the yield point, [[gamma].sub.p] = [[[gamma].sub.p].sup.y]. [[[gamma].sub.p].sup.y] is defined as the rate of plastic deformation at the yield point, and is specified as follows: When yield initiates, the stretch ratio [[a.sub.m].sup.y] is equal to 1 + [[epsilon].sup.y] and [a.sub.m] is equal to zero. Then the second part of Eq 7 is equal to zero, and using the approximations given elsewhere (15), we obtain:

[[[gamma].sup.y].sub.p] [congruent] a/a([[a.sup.y].sub.m] - 1) (12)

where is the rate deformation of the reference length.

At this stage of deformation ([epsilon] = [[epsilon].sup.y]), and because the normal distribution function is symmetric, P[([epsilon]).sup.y] becomes equal to 1/2, and therefore, following Eqs 11 and 12 we obtain:

1/2 k = [[[gamma].sup.y].sub.p] = a/a([[a.sup.y].sub.m] - 1)


[[gamma].sub.p] = a/a([[a.sup.y].sub.m] - 1) 1/s[square root]2[pi] [[[integral].sup.a].sub.1] [e.sup.-1/2[([a.sub.l] - [micro]/s).sup.2] (13)

where the limits of integration have been substituted with the corresponding values of stretch ratio a instead of strain [epsilon]. Parameter [micro], which is the mean value of the probability density, is the stretch ratio [[a.sub.m].sup.y], where yield occurs and was taken experimentally equal to 0.06. The standard deviation s is the only fitting parameter equal to 0.011. Combining Eqs 4, 5, 6 and 13, the stress-strain response of the glassy state at different crosshead speeds can be described while the strain softening is predicted in a self-consistent manner. Integration in Eqs 4 and 13 has been made numerically using small time steps, until a high convergence has been obtained. The set of parameters required is the same as the one obtained from creep experiments. The theoretical results in respect to the experimental data are shown in Fig. 6, where a good approximation has been testified for the entire shape of the experimental curves.

Apart from the model capability to describe the total stress-strain behavior of the glassy polymers, it will now be shown that the complete description of creep procedure may be achieved as well. In this case, the calculation of the plastic part of strain is required, especially for stress levels close to the material yield stress. Therefore, in order to further check the flexibility of the model, creep strain versus time was calculated with an analogous procedure: Eqs 2 and 3 were combined by also taking into account the rate of plastic deformation given by Eq 13. The plastic strain [[epsilon].sub.p] that is developed during creep procedure can then be calculated via an integration of [[gamma].sub.p]. The recoverable strain [[epsilon].sub.R], which is included in Eq 3, is consequently given by:

[[epsilon].sub.R] = [epsilon] - [[epsilon].sub.el] - [[epsilon].sub.p] (14)

where [[epsilon].sub.el] is the instantaneous elastic strain.

The model predictions are plotted in respect to the experimental creep data in Fig. 7, for representative stress levels. It must be noted here that calculations have been based on the same parameter values as mentioned above, indicating the physical meaning of these quantities. The deviation observed at the stage of creep failure is attributed to the geometrical instability (necking) that takes place during tensile creep experiments at high stress levels. Nevertheless, the model is able to predict the gradual acceleration of inelastic strain rate to higher values.

The experimental method that was used permits the detailed description of the deformation distribution along the gauge length. Therefore, it is possible to localize a specific region where the magnitude of strain is much higher then in the other regions. When yielding occurs followed by necking initiation at this specific region, a large deviation of strain appears. From the theoretical point of view, the rate of plastic deformation [[gamma].sub.p] is given by Eq 13. The form of this equation simulates the experimentally observed increment of the rate of deformation in the above-mentioned localized region. Two quantities determine the way this abrupt change of the rate of deformation takes place: The mean value [micro] of strain in this region, and the standard deviation s. The first quantity is defined by the macroscopic yield strain [[a.sup.y].sub.m], which is obtained experimentally. The standard deviation s is a fitting parameter, and in combination with the rest of the model parameters resembles with a certain amount of accuracy the experimental data.


The nonlinear viscoelastic and viscoplastic response of the glassy state is predicted in terms of a unified model. This model treats the initial stages of deformation up to the yielding, as a thermally activated process, while the subsequent plastic path is quantitatively described through a mechanism based on the distributed nature of strain around specific regions with high free volume. The elastic energy stored in these regions attains a critical value, leading to the transition of these regions into a nonrecoverable state, exhibited macroscopically by plastic deformation. It was suggested that the deformation mechanism is the same under a constant crosshead speed experiment, and in a creep test with varying stress levels over a wide range.

The material properties required in the model, and related to thermal activation, were obtained from a set of tensile creep experiments, in terms of creep strain rate versus strain. Moreover, two internal variables describing the distribution of strain, namely the yield strain (mean value of the probability density) and the standard deviation, were also needed for the model prediction of plastic strain rate.

The model, hereafter, was proved to be capable of predicting the nonlinear viscoelastic-viscoplastic response under a nonmonotonic loading test for various crosshead speeds test, as well as the creep strain versus time for various stress levels.


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Publication:Polymer Engineering and Science
Date:Aug 1, 2001
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