A state variable model for high strength polymers.

INTRODUCTION

Polymers are replacing metallic materials in many load-bearing applications. Both single constituent polymers and composites are being used. These polymer components are expected to perform as reliably and predictably as the metallic components they are replacing. At room temperature, polymers can exhibit significant time (rate)-dependent deformation behavior. However, most of the engineering analysis of polymeric parts is done with the time-independent, linear elastic relations. This procedure either requires large safety factors with an inefficient use of the polymeric materials or entails high risk of failure of the components.

Engineers designing metallic components operating at high temperature have ensured reliable service by performing inelastic analysis and life prediction in the design stage long before the component is built. Modern constitutive theories and powerful, economic computation make the use of inelastic analysis in the design office possible. The same method can be applied for polymeric components, provided the proper constitutive equation to describe the time-dependent deformation behavior is available.

Current research on the mechanical behavior of polymeric materials is aimed at elucidating specific properties such as the yield phenomenon, creep, and rate dependence in monotonic loading. Often, one paper deals with one of these subjects, but a follow-up paper on the other properties for the same polymer cannot be found. For engineering applications, the "total time (rate)-dependent behaviors," such as rate sensitivity, creep, relaxation, and cyclic loading, are of importance. Also, nonlinear integral representations are favored in polymer research. A constitutive equation in differential form can be fully equivalent to the integral representation and is computationally advantageous.

The importance of proper constitutive equations in the stress analysis of polymers is beginning to be recognized and important issues have been addressed already, Stokes and Nied (1,2), Nied and Stokes (3), Nied, Stokes and Ysseldyke (4) and Amoedo and Lee (5), to name just a few. The intent of this paper is to contribute to the development of constitutive equations to be used for polymers in engineering analysis.

With our experience in high temperature constitutive equation research for metallic materials, we investigated the deformation behavior of high strength commercial polymers at room temperature. Evaluative, descriptive, and discriminating tests were performed. These tests suggest that a unified, state variable model can be useful in modeling the behavior of these polymers. An overstress dependence of the inelastic rate of deformation can capture both the usual and some anomalous behaviors. On the other hand, features of deformation were found, such as the large recovery and the curved unloading, that emphasize the fact that very different micromechanisms are operative in polymers than in metals at high homologous temperature.

EXPERIMENTAL SETUP AND POLYMERS TESTED

All tests were performed at room temperature using a servohydraulic, computer-controlled, MTS axial/torsion testing machine. A clip-on extensometer, applied to specimens of cylindrical cross section, was used to measure strain. Details are to be found in Bordonaro and Krempl (6). The tests were performed either in strain control, i.e., the axial component of the deformation gradient is enforced, or in stress control, i.e., the load is controlled. None of the test results involves inhomogeneous deformation (necking). We have tested commercially available nylon 66, poly(ether imide) (PEI), which was purchased as Ultem 1000 (GE Plastics), and poly(etherether ketone) (PEEK); see the details in Bordonaro and Krempl (7). Nylon 66 and PEEK are semicrystalline and PEI is an amorphous polymer.

RESULTS

Strain Control

Figure I shows the rate sensitivity during loading and unloading for the three polymers. The region for a common initial slope is highest for nylon 66 and smallest for PEI. There is a nonlinear relation between the flow stress(1) and the strain rate.(2) A hundred-fold increase in the strain rate causes a much less than hundred-fold increase in the flow stress level. Unloading is curved for all three polymers. It was found [[ILLUSTRATION FOR FIGURES 7-9 OMITTED] of Bordonaro and Krempl (7)] that there is considerable strain recovery (sometimes exceeding 50% of the initial strain at zero load) in up to 24 h. After this time, the recovery rate is small and it is unlikely that complete recovery will ever occur. It is not unreasonable to assume that a permanent set can remain.

A decrease in the strain rate by two orders of magnitude at the point of unloading causes an immediate vertical drop. Subsequently, the usual unloading pattern begins; see Figs. 4 through 6 of Bordonaro and Krempl (7). Also, there is an almost elastic behavior upon reloading after a relaxation test; see Figs. 4, 6, and 7 of Bordonaro and Krempl (6). These behaviors suggest that the rate of deformation is the sum of the elastic and inelastic parts.

It is conceivable that nonlinear elasticity may be appropriate for modeling the unloading behavior. Figure 2 shows that this is not the case, however, as a rate increase upon unloading results in a considerably different stress-strain curve. In addition, relaxation is observed to occur during initial loading. Such effects would not be present if the behavior were governed by nonlinear elasticity. Similarly, the behaviors during repeated relaxation tests while unloading and reloading, see Fig. 3, does not support a nonlinear elastic model. All three materials, see Figs. 10-12 of Bordonaro and Krempl (7), exhibit the same pattern of behavior.

The results of the repeated relaxation tests also show an anomalous behavior. Upon unloading, the stress magnitude decreases initially during the relaxation periods, and then starts to increase as unloading progresses! Upon reloading, the stress magnitude increase changes to a stress magnitude decrease as the stress at the beginning of the relaxation test increases.

Bordonaro and Krempl (6) have shown that the relaxation behavior is nearly independent of the strain and stress at which relaxation begins, see their Fig. 5. The relaxation drop in a constant time, however, increases with an increase in prior strain rate. This is true only if the regular flow stress has been reached prior to the start of the relaxation test.(3) Owing to the limited ductility of PEEK and PEI, the regular flow stress cannot be reached, and this property can be demonstrated only for nylon 66. However, nothing in the behavior of PEEK and PEI suggests that they would behave differently if they would allow the reaching of the flow stress before fracture occurs.

Implications for Modeling

Some experiments indicate clearly that nonlinear elasticity is not appropriate. Others demonstrate that there are some aspects of the mechanical behavior of polymers that are similar to those of metals. The initial elastic slope, the elastic reloading after a relaxation test, and the independence of the relaxation behavior on initial stress and strain are also found in metallic materials; see Krempl and Kallianpur (8) and Majors and Krempl (9). However, the curved unloading [ILLUSTRATION FOR FIGURE 1 OMITTED], the large recovery, and the merging of the stress-strain curves of nylon 66 at large strains [ILLUSTRATION FOR FIGURE 1 OMITTED] are properties that are not in common with metals. The phenomena are found in polymers at comparatively large strains. Consequently, a finite deformation theory is appropriate.

The test results, combined with our experience with the modeling of the deformation behavior of metallic materials, strongly suggest

D = [D.sup.el] + [D.sup.in] (1)

where [Mathematical Expression Omitted], with [C.sup.-1] the elastic compliance matrix and [Sigma] the Cauchy stress. A superposed denotes an objective derivative. D is the rate of deformation tensor. The relaxation properties imply that

[D.sup.in] = F[[Sigma] - g] (2)

where g is the equilibrium or back stress, a state variable of the theory, for which a growth law must be specified. This law will be left unspecified in this paper. The function F is increasing and F[0] = 0.

We note that for polymers, tr F [not equal to] 0 since the inelastic deformation is sensitive to superposed hydrostatic pressure; see Stemstein, Ongchin, and Silverman (10), Sternstein and Ongchin (11), Mears, Pae and Sauer (12), Pae (13), and Spitzig and Richmond (14).

For the uniaxial case, Eqs 1 and 2 reduce to

[Mathematical Expression Omitted]

To see that such an equation can reproduce the observed behavior in a qualitative way, we presume that an evolution equation for the equilibrium stress, g, can be found so that the overstress, [Sigma] - g, has the desired property.

The change from stress magnitude decrease to stress magnitude increase during the relaxation periods shown in Fig. 3 can be reproduced if g evolves inside the stress-strain hysteresis loop. This is demonstrated with numerical experiments in Fig. 4. The experimental results can be reproduced qualitatively with an evolution law that is still under development.

To show that the model can predict the independence of the relaxation behavior on location, we specialize Eq 3 for relaxation, D = 0, to obtain

[Mathematical Expression Omitted]

Clearly the stress rate depends only on overstress. The overstress present at the start of the relaxation test is determined by the solution of Eq 3, which can be rewritten as

[Mathematical Expression Omitted]

The stationary solution for the overstress for constant D, indicated by { }, is

[Mathematical Expression Omitted]

The stationary solution can be obtained when the right-hand side is constant. This is possible when g has a constant slope. If the asymptotic solution is reached prior to the beginning of each of the relaxation tests, then the relaxation drop will be independent of location. It is plausible that this was the case for the experiments in question.(4) It should also be noted that since the overstress increases with the prior rate of deformation, the initial relaxation rate also increases. Consequently, the overstress concept can, in principle, reproduce the experimental results of Fig. 5 of Bordonaro and Krempl (6).

The results of the strain controlled tests discussed above seem to be consistent with the predictions from the overstress model. We now proceed to examine experiments that were run in load control as an additional test for the applicability of the theory. These tests were conceived when the state variable model for metals, the viscoplasticity theory based on overstress (VBO), was under development.

Load Controlled Tests

Figure 5a shows the results of a load controlled test with creep periods of 700 s for nylon 66. Upon loading, creep develops gradually as the stress level of the creep test increases. At the highest stress, creep was very rapid and the test had to be terminated after 71 s. Upon unloading, the first creep stress level shows positive creep. The subsequent creep periods show a decrease in the strain that becomes larger as the creep stress level decreases to zero stress. The behavior of PEI, Fig. 5b, is qualitatively the same. In this case, however, the negative creep is highest at 56.6 MPa. As the stress level decreases from this point, the average creep rate magnitude decreases. This behavior is different from nylon 66 where the creep rate magnitude increases steadily with a decrease in stress. It is also interesting to observe that for both polymers, at a given stress level, reverse creep is quite often more pronounced than creep during loading. The creep curves of Fig. 5c shows that all creep is primary. This is also the case for nylon 66.

Evidently, at the same creep stress level, the creep rate can be different on loading and on unloading. Conventional creep theory assumes that the creep rate depends on both the stress and either the creep strain (strain hardening theory) or the creep time (time hardening theory). It therefore predicts that the creep strain rate is the same whenever the stress level is the same. This is not consistent with the observed behavior.

Figure 6 shows results from a numerical experiment that uses the overstress concept. The curves are obtained from a working model that is still under development. The simulation follows the test that was shown in Fig. 5a, and it can be seen that the model reproduces the results in a qualitative fashion. At the same stress level, the creep rate is different on loading than on unloading. The sign change of the overstress is responsible for modeling this behavior. The overstress is positive during loading and becomes negative for the unloading portion of the stress-strain curve.

Figure 7 includes test results for nylon 66, which show that the creep rate need not increase with an increase in stress level. In Fig. 7a the stress-strain curve is shown; the creep curves are plotted in Fig. 7b. Despite an increase in stress, the creep rate decreases. Again, this behavior is considered an anomaly, but it follows directly once the overstress dependence of the inelastic rate of deformation is introduced.

Figure 8 shows results from another numerical experiment. This experiment was patterned after the test shown in Fig. 7. Notice that the model is able to simulate a decrease in the creep rate with an increase in the stress. This is accomplished through the overstress dependence of the inelastic rate of deformation. The overstress at the start of the second creep period is smaller than at the start of the first creep period. This produces the decrease in the inelastic rate of deformation, or the creep rate, as the stress increases.

DISCUSSION

In the above, we have presented evidence that non-linear elasticity is not a proper model for the three polymers. On the other hand, experimental results demonstrate the usefulness of a state variable model where the inelastic strain rate depends on the overstress. Indeed, these tests are considered discriminating and were invented when the viscoplasticity model for metals (VBO) was under development; see Krempl (15), Kujawski, Kallianpur, and Krempl (16), Krempl (17), Krempl and Kallianpur (18), and Krempl (19). We again introduced them for the polymers without knowing the outcome. They confirmed the usefulness of the overstress concept in modeling inelastic deformation. In many aspects, the results are qualitatively similar to those of metals. For this reason, we embarked on developing a state variable model based on overstress.

Of course, there may be other models that can simulate the results discussed above. At this time, however, we are unaware of any. The overstress concept has been found useful by Kitagawa, Mori, and Matsutani (20) and Kitagawa and Takagi (21) for the modeling of polypropylene. They used the deformation theory based on overstress, which is not applicable for cyclic loading. Consequently, modeling was quite good for monotonic loading but not acceptable for cyclic loading. Hasan and Boyce (22) also use the rate of deformation to be the sum of elastic and inelastic contributions. The dependence of the inelastic strain rate on stress and temperature is derived from micromechanical arguments. G'Sell and Jonas (23) discuss an overstress-like idea in the analysis of their test data. They do not develop a model in this paper.

This approach differs from both the customary integral models, see O'Dowd and Knauss (24), and the network models, see Boyce and Arruda (25), that are common in the polymer community.

[TABULAR DATA OMITTED]

Also, the present approach differs from what one might call the "static internal stress concept" that can be found in, for example, Kubitt, Petermann, and Rigdahl (26). As can be seen from the numerical experiments in Figs. 4, 6, and 8, the equilibrium stress, which would be equivalent to the internal stress, evolves with deformation. It is therefore very difficult to assign a particular value for the equilibrium stress to a material as is done in the paper cited. However, we share the view that the inelastic strain rate depends on the overstress or effective stress.

We advocate the use of the overstress concept within a state variable model for both metals and polymers. Such an approach seems to ignore the fundamental differences of the mechanisms of inelastic deformation in metals and polymers. Dislocation movement, change of dislocation density, and change of the defect structure with deformation are strong mechanisms in metallic materials. In amorphous polymers, the uncoiling of chain molecules leads to inelastic deformation, and in semicrystalline polymers, deformation mechanisms include the stretching of tie molecules, and changes in the interlamellar spacing. We are not saying that the same model is applicable for both materials; we are only postulating that the basic structure and the use of the overstress and of the equilibrium stress are common. It is seen from Eq 3 that in equilibrium, when all rates are zero, the stress is equal to the equilibrium stress. This stress is a measure of the internal makeup and of the elastic behavior. It is conceivable that a material with a high density of obstacles to inelastic deformation can support a greater stress than a material with a low density. Obstacles are present in both types of materials. When the deformation is away from equilibrium, the equilibrium stress evolves and continues to be a measure of the defects and obstacles in the material. It is, however, continuously changing with deformation. Because of this property, and because it takes a very long time to reach equilibrium, the equilibrium stress cannot be measured directly.

In other aspects, metals and polymers are quite different. At pressures below the ultimate strength of a metallic material, inelastic deformation is nearly unaffected by superposed pressure. This is not the case for polymers. Other differences include:

We proceed on the assumption that the differences in the mechanical behavior can be accommodated within a state variable model by postulating different growth laws for the state variables for metals and polymers. These growth laws bring out the differences between the two classes of materials while maintaining the overstress concept. Also, the formulation will be in the form of differential equations that are computationally advantageous. Work continues to achieve this goal.

The evolution laws must be specific to a given material since differences exist even between the polymers. In nylon 66, the negative creep rate and the positive relaxation rate increase steadily during unloading; see Fig. 10, of Bordonaro and Krempl (7) and Fig. 5a. For PEI, on the other hand, a maximum creep or relaxation rate is reached at some intermediate stress level; see Fig. 12 of Bordonaro and Krempl (7) and Fig. 5b. These results suggest that the evolution law for the equilibrium stress cannot be the same for both polymers.

It is also interesting that at the level of our testing, we would not have been in a position to ascertain that PEI is an amorphous polymer whereas the other two materials are semicrystalline. Their behaviors are qualitatively the same. A slight difference exists in the repeated relaxation tests during unloading; see Figs. 10 through 12 of Bordonaro and Krempl (7). In nylon 66 and PEEK, the increase in the stress during these relaxation periods becomes larger as the stress level decreases to zero stress. For PEI, the maximum stress increase appears to be at an intermediate stress level. This observation is consistent with the creep behavior mentioned above. These differences are not significant for modeling and would not have given us a hint about the difference in the internal makeup of the polymers.

ACKNOWLEDGMENT 

The financial support of the National Science Foundation is gratefully acknowledged. The paper was written while EK was a Senior Humbolt Scientist at TH Darmstadt, Maschinenelemente und Maschinenakustik, Professor F. G. Kollman, Darmstadt, Germany.

NOMENCLATURE 

D = Rate of deformation tensor.

[C.sup.-1] = Elastic compliance tensor.

[Mathematical Expression Omitted] = Objective rate of Cauchy stress tensor.

[Sigma] = Cauchy stress tensor.

g = Cauchy equilibrium stress.

F = Odd tensor valued function.

[Sigma] - g = Overstress.

1 Flow stress denotes the stress in the region where the modulus is much smaller than the modulus at the origin.

2 Strain rate equals the rate of the axial component of the deformation gradient.

3 See the comments regarding Figs. 4, 5, 7, and 8 in Bordonaro and Krempl (6).

4 Using the definition of D in terms of 1 + [Epsilon], the axial component of the deformation gradient, and its time derivative. [Mathematical Expression Omitted], the right-hand side of Eq 6 can be written as [Mathematical Expression Omitted]. We see that for the asymptotic solution to hold exactly, the expression in parentheses must be constant, since we kept [Mathematical Expression Omitted] constant during the test. This requirement is not in conflict with the experimental results.

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