# An Experimental Investigation of Normal and Shear Stress Interaction of an Epoxy Resin and Model Predictions.

INTRODUCTION

Constitutive modeling of the deformation behavior of solid polymers has attracted considerable interest recently, due to the use of polymeric materials in primary structures. To ensure reliable and safe performance of these structures, stress analysis must be performed, which requires that accurate constitutive equations be available. However, such is not the case.

The choice or development of an appropriate model to predict the mechanical behavior usually necessitates a general understanding of the properties of the material involved, and these properties are mostly found through experiments. The low modulus and sensitivity to the environment of polymeric materials make the experimental investigation of polymers a fairly challenging task, especially when multiaxial loading is applied.

In plasticity, axial-torsional tests are often used to study the interaction between normal and shear stress (strain) components. These tests are employed to discriminate among different constitutive models of plastic deformation. An appropriate constitutive model for polymers should also be able to describe the mechanical response of axial-torsional tests. In polymeric composite structures where polymers are used as matrix materials, the stress state in the matrix is generally of a multiaxial nature. Therefore, the interaction between different stress (strain) components is frequently encountered in these structures. Research on interaction of normal-shear stress (strain) components is an important first step toward modeling more complex stress/strain interaction problems encountered in fiber-reinforced composites.

There are a few experimental investigations reported in the literature for polymers subjected to combined normal and shear stresses. A sample of representative studies is described below. Ewing et al. (1) investigated the combined tension-torsion creep behavior of polyethylene with emphasis on the effect of different loading paths on creep response. The interaction between torsion and axial force or volume dilatation was studied by McKenna and co-workers (2. 3). It was found that the torsion influences the axial force and the volume change of polymers like polymethyl methacrylate (PMMA) (2) and epoxy glass (3). Lu and Knauss (4) investigated the nonlinear thermo-mechanical behavior of PMMA under combined axial (tension, compression) and shear (torsional) stress states. It was shown that dilatation affects the shear creep behavior at moderate strains. The above tests, among others, mainly focused on the creep or stress relaxation behavior of polymers. The transient behavior of polyethylene under different stra in histories was investigated by Kitagawa and Takagi (5). Krempl and Bordonaro (6) performed non-proportional loading tests on nylon 66 at room temperature. To the best of our knowledge, no tests have been performed to study the axial-torsional interaction behavior of epoxy resins in which the test temperature is far below the glass transition temperature.

Epoxy resins have certain properties that are different from those of thermoplastics and other unsaturated thermosets. Notable among these is much lower strains to failure. Epoxy resins are widely used in fiber-reinforced composites as matrix materials, and consequently, a thorough understanding of their nonlinear behavior is a prerequisite for the accurate prediction of the mechanical response of epoxy-based composites and design of such composite structures.

This investigation focuses on the axial-torsional interaction effect of an epoxy resin. Thin-walled tubular specimens are subjected to combined constant normal (or shear) stress and varying shear (or normal) stress to study the creep behavior. Different strain paths at constant octahedral shear strain-rate are also applied to the tubular specimens to investigate the transient stress responses. In addition, different constant octahedral shear strain-rates are used to study the rate effects. Furthermore, the normal-shear stress interaction is investigated by creep tests with non-proportional stress histories. It is shown that superimposed normal stress influences the shear strain response and vice versa. Two typical viscoelastic constitutive models are chosen to check their capability to predict the above observed experimental characteristics of the epoxy resin.

SPECIMEN PREPARATION AND TEST FACILITY

Specimen Fabrication

The material used in the experimental investigation was Epon 828, an epoxy resin from the reaction product of epichiorohydrin and bisphenol A, commercially known as ColdCure epoxy. The hardener is a methylenebiscyclohexylamine. The ColdCure epoxy resin and hardener were mixed with 2:1 (volume) ratio and stirred thoroughly, and then poured into a tubular plastic mold. The samples were cured for 1 hour at 65[degrees]C and cooled to room temperature in the oven. Following this, they were held at the ambient temperature for 7 days to reach 100% polymerization prior to machining. Test specimens were manufactured from the cured samples by a computer numerical controlled (CNC) lathe that provides for precise machining of any transition profile geometry. Aluminum end tabs were glued to the tube ends (3-M Type DP-460 Epoxy Adhesive) so that the specimen can be inserted into the gripping system of the test machine. In this way one avoids damaging the specimen extremities during the gripping process. The aluminum end ca ps are also needed because the contact force between the gripping system and tube specimen will relax with time, if the epoxy tube ends are gripped directly. This will then cause the specimen to slip out of the gripping system during tensile tests, and for the oil to leak during differential pressurization (biaxial tests). The final specimen configuration is shown in Fig. 1.

Test Facility

All tests were performed in a servo-hydraulic, computer-controlled, multiaxial testing machine described in ref. (7). This test facility can apply a combination of axial load, differential pressure (inside and outside pressures) and torsion, and is capable of generating all principal stress and strain ratios, as well as change in the direction of the applied principal stresses and strains. For the current investigation, the axial and torsional loading systems were used, except for a shear creep in which a superimposed hydrostatic pressure was applied to the specimen.

A strain gauge in the axial direction served as an axial strain transducer, while a pair of strain gauges mounted at [+ or -] 45[degrees] to the axial direction served as a shear strain transducer. Another pair, mounted on the opposite surface of the tubular specimen (Fig. 2), was used as a supplement to record strains in the [+ or -] 45[degrees] directions. The recorded data from the latter strain gauges served two main purposes. The first was to record the tensile and compressive strains under a pure shear stress. The epoxy resin has been shown to be sensitive to hydrostatic stress, e.g. see ref. (8), these results, therefore, can be used as a base data to compare the effect of a superimposed hydrostatic stress. The second purpose was to confirm that the specimen was aligned coaxially with the grip system. The strain ranges for the axial and shear strain gauges were all calibrated to [+ or -] 5% strain with a resolution of 0.0005%.

The axial and shear stresses imposed on a thin-walled tube subject to an axial load, F, and a torque, T, can be calculated from:

[[sigma].sub.a] = 4F/[pi]([D.sup.2]-[d.sup.2]) (1)

[tau] = 16DT/[pi]([D.sup.4]-[d.sup.4]) (2)

where D and d are the outer and inner diameters of the tube, and [[sigma].sub.a] and [tau] and are the axial and shear stresses, respectively.

The shear stress, [tau], is the stress value on the outer surface of the tube. It is calculated based on the assumption of linear strain distribution across the wall thickness. By using a linear stress variation, the deviation from the average value was [+ or -] 3.96% for the current specimen geometry.

The octahedral shear stress, [[tau].sub.oct], and hydrostatic stress, [[sigma].sub.h], are defined as:

[[tau].sub.oct] = [square root of (2)]/3 [square root of ([[sigma].sup.2.sub.a] + 3[[tau].sup.2])] (3)

[[sigma].sub.h] = [[sigma].sub.a]/3 (without superimposed hydrostatic pressure p) (4a)

or

[[sigma].sub.h] = [[sigma].sub.a]/3 - p

(under superimposed hydrostatic pressure p) (4b)

TEST PROCEDURE

Creep Tests

a) Axial stress effect on shear creep

Two different shear creep tests were conducted (Series I, see Table 1): one with a superimposed constant axial tensile stress (Case 1) and another with a superimposed constant axial compressive stress (Case 2). Both tests have the same axial stress value of 14 MPa and shear stress value of 10 MPa, but the sign of the axial stress varies. These tests were used to investigate the influence of the normal stress on the shear creep behavior.

b) Hydrostatic pressure effect on shear creep

This series of tests included two types of shear creep tests. Here a uniform hydrostatic pressure was superimposed on the shear stress, rather than the axial tensile and compressive stresses as used in the previous test series. The applied shear stress and hydrostatic pressure were 15MPa and OMPa (no hydrostatic pressure) for Case 1, see Table 1, and 15MPa and 17MPa, respectively, for another test (Series II, Case 2). These tests will indicate the effect of the hydrostatic pressure on the shear creep behavior.

c) Shear stress effect on axial creep

The third series of creep tests consisted of an axial tensile test with a superimposed shear stress (Series III in Table 1). In these tests the axial stress value was 14 MPa but the magnitude of the shear stress differed. In this manner the influence of the shear stress on the normal creep behavior can be investigated.

The stresses indicated in Table 1 (except the hydrostatic pressure) were applied within 10 seconds and subsequently held constant for a period of one hour. The resulting creep curves were analyzed for periods beyond 20 seconds, to allow transient loading effects to diminish. For the test with a superimposed hydrostatic pressure, the pressure was applied prior to starting the shear creep test.

Non-Proportional Path

a) Strain-controlled tests

A non-proportional strain-controlled loading scheme is shown in Fig. 3. Two different paths terminate at the same point C in the [[epsilon].sub.a] vs. [gamma] strain space. The testing procedure consisted of loading to a certain axial (shear) strain value that was then held constant. Subsequently, the shear (axial) strain was increased to the coordinate of point C. This is the so-called corner test which, as pointed out by Krempl and Bordonaro (6), is the test often used to discriminate plastic deformation behavior.

All strain paths were loaded at the same strain rate. The rate is defined by the octahedral shear strain rate, [[epsilon].sub.oct] = [square root of (2/9 [(1 + v).sup.2] [[epsilon].sup.2.sub.a] + 1/6 [[gamma].sup.2])], where [[epsilon].sub.a] and [gamma] are axial and shear strains, respectively, a superposed dot designates the time derivative, and v is the Poisson's ratio. The corresponding octahedral shear strain is defined by [[epsilon].sub.oct] = [square root of (2/9 [(1 + v).sup.2] [[epsilon].sup.2.sub.a] + 1/6 [[gamma].sup.2])]. For the corner path test, an octahedral shear strain rate of [10.sup.-4] [s.sup.-1] was chosen, which converts to an axial strain rate of 1.52 X [10.sup.-4] [s.sup.-1] and a shear strain rate of 1.22 x [10.sup.-4] [s.sup.-1]. In Fig. 3, the strain components and the octahedral shear strains, in parentheses, are indicated at points A, B and C. In the calculation of the corresponding axial and shear strains and the strain rates, a value of 0.4 for the Poisson's ratio was used.

b) Strain-rate effect

The effect of strain rate was studied by performing the above tests at a different octahedral shear strain rate of [10.sup.-5] [s.sup.-1]. The corresponding axial and shear strain rates were 1.52 X [10.sup.-5] [s.sup.-1] and 1.22 X [10.sup.-5] [s.sup.-1], respectively, for the corner path tests.

c) Stress-controlled tests

Another set of tests was performed to gain a further insight into the axial-torsional interaction. In these experiments the creep behavior of the epoxy resin was investigated when subjected to different non-proportional stress histories. The loading path was the same type of corner test shown in Fig. 3, but the tests were conducted under a stress control mode, instead of the strain control as depicted in Fig. 3. After reaching point C by different stress paths, the axial and shear stresses were kept constant for an hour. The corresponding axial and shear creep deformations were recorded.

The stress rate for the two stress paths was the same. An effective stress rate defined by [[sigma].sub.eff] = [square root of ([[sigma].sup.2.sub.a] + 3[[tau].sup.2])] was used, where [[sigma].sub.a] and [tau] are axial and shear stress rates, respectively. The effective stress rate was chosen to be 0.1 MPa/s, which corresponds to an axial stress rate of 0.1 MPa/s and a shear stress rate of 0.058 MPa/s. The stress values reached at point C were a combination of 21MPa in axial and 15MPa in shear.

All tests were performed at ambient environments (25[degrees]C and 50% humidity). In order to minimize specimen-to-specimen variations, a single specimen was used as much as possible. This required that the specimen deformation from previous tests be recovered. It was found that as long as the applied stress level was not too high and enough time was allowed for the recovery process, the specimen did indeed recover completely. Confirmatory tests were conducted to verify the observed trends and response values.

RESULTS

Creep Tests

a) Axial stress effect an shear creep

Shear creep responses at two different combinations of axial and shear stresses (see Table 1, series I) are shown in Fig. 4. This figure illustrates the influence of the axial (tensile or compressive) stress on the shear creep behavior of the epoxy resin. It is clear that superimposed axial tension and compression have a readily discernible and different effect on the creep behavior in shear. A shear stress with a superimposed axial tensile stress, case 1, exhibits higher creep deformation than that with the same shear stress but a superimposed axial compressive stress, case 2.

The above results can be explained in terms of the hydrostatic stress states. In the case of a shear stress with a superimposed axial tensile stress, the hydrostatic stress is positive. When the axial stress is compressive, the hydrostatic stress is negative. Therefore, a superimposed positive hydrostatic stress accelerates the shear creep deformation, whereas a superimposed negative hydrostatic stress retards the shear creep deformation.

b) Hydrostatic pressure effect on shear creep

As shown in Fig. 4. a shear stress with a superimposed axial tensile stress, case 1. exhibits a higher creep deformation than that with the same shear stress but a superimposed axial compressive stress, case 2, and this difference can be attributed to the hydrostatic stress effect. However, in the above tests, the hydrostatic stress is only one-third of the axial stress; therefore, the hydrostatic stress values are relatively small and, thus, the difference in the creep strain response is not significant. To further explore the hydrostatic stress effect, two shear creep tests, one with a superimposed hydrostatic pressure and another one without it, were conducted and the shear strain responses are shown in Fig. 5. Here, the difference is quite noticeable, i.e., a shear stress with a superimposed hydrostatic pressure retards the creep deformation.

c) Shear stress effect on axial creep

Figure 6 depicts the influence of a superimposed shear stress on the axial tensile creep behavior. It can be seen that a superimposed shear stress has a significant effect on the axial creep response. A superimposed shear stress accelerates the creep deformation. This can also be attributed to the increase in the second stress invariant. A superimposed shear stress increases the effective (or octahedral shear) stress, and the creep deformation accelerates under a higher effective (or octahedral shear) stress.

Non-Proportional Path Tests

a) Strain-controlled tests

The axial stress responses for the two loading paths OAC and OBC at an octahedral shear strain rate of [10.sup.-4] [s.sup.-1] are shown in Fig. 7. The solid square symbols represent the stress response for the loading path OAC and the hollow ones for the loading path OBC. For both strain paths, axial stress increases with the increase of strain until a strain value of 0.742% is reached. Prior to reaching the aforementioned axial strain, the axial stress response for path OBC is slightly below that of path OAC. This difference can only be attributed to the presence of a prior shear stress for the path OBC. Therefore, this figure shows that the presence of a prior shear stress causes a reduction in the stiffness of the axial stress-strain behavior.

The shear response for the same strain paths is depicted in Fig. 8. Again two distinct curves are observed, but the trend is reversed, i.e., the shear stress-strain curve for the path OBC is slight higher than that of OAC because of the presence of a prior axial stress in the latter path. Therefore, the existence of a prior axial stress also causes a reduction in the stiffness of the shear stress-strain curve.

With reference to Fig. 7 and path OAC, the axial stress drops from point A to the lowest point C by about 3.06 MPa when the shear stress, path AC, is applied. The corresponding drop in the shear stress from point B to lowest point C in Fig. 8 is about 1.32 MPa. In Figs. 7 and 8 there are two points labeled C. They are referred to the axial (shear) stresses reached by the different strain paths. The axial and shear stress drops observed in Figs. 7 and 8 are due to the stress relaxation while the axial or shear strains are kept constant. There may also be an interaction influence. In order to determine the contribution to the total stress drop by each of the above factors, another test was performed in which the axial strain was first applied at a constant octahedral shear strain rate of [10.sup.-4] [s.sup.-1]. After it reached point A, the strain was kept constant instead of applying a shear strain as in path OAC of Fig. 3. The corresponding stress response (OAC') is shown in Fig. 9, in conjunction with that o f path OAC. Note that the stress drop from A to C' is due to stress relaxation only, while the stress drop from A to C is a result of both the stress relaxation and shear stress interaction. It is seen from the figure that about 1/3 of the total stress drop can be attributed to the shear stress interaction effect.

b) Strain-rate effect

Figures 10 and 11 show the axial stress versus axial strain and the shear stress versus shear strain for two octahedral shear strain rates of [10.sup.-4] [s.sup.-1] and [10.sup.-5] [s.sup.-1]. It is noted that there is no significant relative difference between the stress responses due to the path effect as a result of change in the strain rate of one order of magnitude. However, the change in the strain rate has a noticeable influence on the stiffness of the stress-strain response, i.e., the elastic modulus is a function of time.

c) Stress-controlled tests

The axial and shear strain responses for a corner test with a maximum axial stress of 21 MPa and a maximum shear stress of 15 MPa, are shown in Figs. 12 and 13, respectively. The figures show different strain responses for the two loading paths, with the creep deformation from path OBO being higher than that from path OAC for both the axial and shear strain components.

The experimental results discussed herein clearly show that the stress (strain) response of the epoxy resin is stress or strain component dependent, that is, an axial stress (strain) will influence the shear strain (stress) response, and vice versa. The influence is observed in both the transient and creep responses.

CONSTITUTIVE MODELING

A number of constitutive models have been proposed to predict the nonlinear viscoelastic response. For example, the free volume theory of Knauss and Emri (9) and Lou and Schapery thermodynamic based model (10) are among those well-known and often quoted. This type of integral representation is convenient for the stress and strain analysis, and the experimental data required for the determination of material constants is not onerous. However, their application in the multiaxial stress states has not been fully explored (11).

There is no interaction between the axial stress (strain) and the shear stress (strain) in the linear range of a viscoelastic material; therefore, axial-shear interaction is a nonlinear phenomenon. In certain cases, nonlinear viscoelastic models are obtained by modifying linear viscoelastic models by replacing ordinary tune in the kernel functions with a strain (stress) modified time. Collectively these models are known as "strain clock" and "stress clock" models depending upon whether the modified time is made to depend on the strain or stress, respectively. The free volume model developed by Knauss and Emri (9) is a case in point, in which the strain clock depends upon the first strain invariant, and this dependence is motivated by the free volume consideration. As pointed out by Krempl and Bordonaro (6), such a model would not be able to reproduce the results reported here because a prior torsion has no influence on a subsequent tension loading in this model.

A simple way to predict coupling between normal and shear stresses of isotropic materials in small deformation would be to include a second invariant although this is not necessarily the only form. An attempt was made by Popelar and Liechti (12) to formulate such a model. Although originally it was proposed to consider the nonlinearity for shear-dominated loadings, it may also be able to predict the axial-shear interaction because of the inclusion of the effective strain in the reduced time (intrinsic time). However, little experimental verification has been carried out on this model, and its predictive capabilities must be further explored.

The Schapery model (10) can, in principle, account for the torsion-axial coupling because the equivalent stress can be included in the nonlinear functions, as demonstrated by Zhang et al. (13). In their paper, a set of combined torsion-axial creep tests was performed, and a good agreement between test results and model predictions was obtained. The equivalent stress, [[sigma].sub.eq], initially defined in the above constitutive model accounted for the distortion energy only, i.e., it was the von Mises equivalent stress. However, as shown in our creep tests and others, see e.g. (4, 14), hydrostatic stress has a discernable effect on the shear creep response of polymers. The hydrostatic stress is related to the bulk dilatation energy. If only an equivalent stress based on the von Mises is used (distortion energy), then the hydrostatic stress influence is not accounted for. Therefore, the equivalent stress, [[sigma].sub.eq], has to be modified for polymers. Lai and Bakker (15) have attempted to include the hydr ostatic stress influence in a modified Schapery model. However, it has been shown (11) and will be shown later that the model does not adequately simulate the hydrostatic stress influence on the mechanical behavior of epoxy resins.

To illustrate the predictive capability of the current constitutive models, two typical nonlinear viscoelastic models, one by Lai and Bakker (15), a modified Schapery model in the integral form, and the other by Xia and Ellyin (16), a differential form model, are chosen to predict the mechanical response of the epoxy resin under the aforementioned loading paths. The equations for the two nonlinear viscoelastic constitutive models are summarized below.

Nonlinear Constitutive Models

a) Integral representation

For a thermo-rheologically simple isotropic viscoelastic material under a multiaxial state of stress, the constitutive law can be expressed as:

[[epsilon].sub.ij] = 1/2 [J.sub.0][g.sub.0][s.sub.ij](t) + 1/2 [g.sub.1] [[integral].sup.t.sub.0] [DELTA]J[[psi](t) - [psi]'([tau])] d([g.sub.2][s.sub.ij])/d[tau] d[tau] + 1/9 [B.sub.0][g.sub.0][[delta].sub.ij][[sigma].sub.kk](t) + 1/9 [[delta].sub.ij][g.sub.1] [[integral].sup.t.sub.0] [DELTA]B[[psi](t) - [psi]'([tau])] d([g.sub.2][[sigma].sub.kk])/d[tau] d[tau] (5)

where [J.sub.0] and [B.sub.0] are instantaneous shear and bulk compliances; [DELTA]J and [DELTA]B are transient shear and bulk compliances, respectively; [psi] and [psi]' are reduced times,

[psi] = [[integral].sup.t.sub.0] d[tau]/[a.sub.[sigma]] and [psi]' = [[integral].sup.[tau].sub.0] d[tau]/[a.sub.[sigma]];and [g.sub.0] and [a.sub.[sigma]] are nonlinear functions of [[sigma].sub.kk], the first invariant of a stress tensor; [g.sub.1] and [g.sub.2] are nonlinear functions of the von Mises equivalent stress.

b) Differential representation

In this model, the total strain rate {[[epsilon].sub.t]} is assumed to be the sum of the elastic and creep strain rates, {[[epsilon].sub.e]} and {[[epsilon].sub.c]}, respectively, i.e.,

{[[epsilon].sub.t]} = {[[epsilon].sub.e]} + {[[epsilon].sub.c]} (6)

The elastic strain rate is calculated through Hooke's law,

{[sigma]} = E[[A].sup.-1]{[[epsilon].sub.e]} (7)

For a number of Kelvin (Voigt) elements connected in series, creep strain rate {[[epsilon].sub.c]} is the sum of the strain rate of each element {[[epsilon].sub.ci]}, which is nonlinearly related to applied stress {[sigma]},

{[[epsilon].sub.c]} = [summation over (n/i=1)]{[[epsilon].sub.ci]} = [summation over (n/i=1)]([a.sub.i][A][[sigma].sup.[[alpha].sub.i]-1.sub.eqM]{[sigma]} - [b.sub.i]{[[epsilon].sub.ci]}) (8)

where [a.sub.i], [b.sub.i] and [[alpha].sub.i] (i = 1, 2 ..., n) are material constants, E is Young's modulus, [A] is a matrix related to the Poisson's ratio, [[sigma].sub.eqM] is a modified von Mises equivalent stress, which includes the hydrostatic stress term (17).

All the model constants and functions for the integral representation are calibrated from a set of uniaxial creep-recovery curves at different stress levels (11). The stress in creep tests varied from 2.8 MPa to 23.4 MPa. The adopted method to determine functional forms and constants was that advocated by Schapery (18) and widely used by other researchers, see e.g., Peretz and Weitsman (19). The basic characterization procedure involves fitting the uniaxial form of Eq 5 to linear creep-recovery curve at low stress level, as a first step, to determine [J.sub.0], [B.sub.0], [DELTA]J and [DELTA]B. Then the uniaxial integral equation is fitted with each creep-recovery curve and the stress dependence of [g.sub.0], [g.sub.1], [g.sub.2] and [a.sub.[sigma]] is determined. The function relations between [g.sub.0], [g.sub.1], [g.sub.2], [a.sub.[sigma]] and the applied stresses are finally fitted by least square method. These model constants and functions for the integral representation are listed in Table 2.

In order to compare the prediction capability of these two models, the same creep-recovery curves were used to obtain the material constants for the differential model. The uniaxial form of Eq 8 indicates that the strain recovery depends only on the constants [b.sub.1] and [b.sub.2] ([sigma] = 0), therefore the two material constants can be determined first from the strain recovery curve at a given stress level (16). Although any strain recovery curve can be used for the calibration of [b.sub.1] and [b.sub.2], the average values of [b.sub.1] and [b.sub.2] from all stress levels were used here. Subsequently, the constants E, [a.sub.1], [a.sub.2], [[alpha].sub.1] and [[alpha].sub.2] are determined based on the best fit to the creep curves as follow:

For a constant stress [[sigma].sub.0], the uniaxial creep strain can be reformulated from the above differential relations as

[epsilon] = [[sigma].sub.0]/E + [[sigma].sup.[[alpha].sub.1].sub.0]/[E.sub.1] (1 - [e.sup.-[b.sub.1]t]) + [[sigma].sup.[[alpha].sub.2].sub.0]/[E.sub.2] (1 - [e.sup.-[b.sub.2]t]) (9)

where [E.sub.i] = [b.sub.i]/[a.sub.i] (i = 1, 2). Fitting the above equation with the experimental creep curve at the stress level [[sigma].sub.0], the values for E, [[sigma].sup.[[alpha].sub.1].sub.0]/[E.sub.1] and [[alpha].sup.[[alpha].sub.2].sub.0]/[E.sub.2] can be obtained. When fitting Eq 9 with all the creep-recovery curves at different stress levels, two curves, [[sigma].sup.[[alpha].sub.1]]/[E.sub.1] ~ [sigma] and [[sigma].sup.[[alpha].sub.2]]/[E.sub.2] ~ [sigma], are obtained. Therefore, [[alpha].sub.1], [E.sub.1], [[alpha].sub.2] and [E.sub.2] can be determined by a least-square fitting of these two curves. Thus, the material constants for the differential representation are found to be (11):

[a.sub.1] = 3.4 X [10.sup.-8] [(MPa).sup.-1.47], [[alpha].sub.1] = 1.47, [b.sub.1] = 0.001,

[a.sub.2] = 2.70 X [10.sup.-12] [(MPa).sup.-4.38],

[[alpha].sub.2] = 4.38, [b.sub.2] = 1 X [10.sup.-4],

E = 2450 MPa and v = 0.4

which represents two Kelvin elements and one elastic element in series in a uniaxial stress representation.

COMPARISON OF MODEL PREDICTIONS WITH EXPERIMENTAL DATA

Creep Teats

The predicted shear and axial creep deformations by the two models are shown in Figs. 4-6. It is noted that the predictions for the uniaxial tensile stress state (series III, case 2) are in good agreement with the experiment data, Fig. 6. This is to be expected since the parameters for the two viscoelastic constitutive models were fitted to the uniaxial tensile creep curves. However, for all other stress states, the predicted creep curves generally overestimate the experimental results. The predictions are in qualitative agreement with the experimental results, noting that the two models do predict some interaction between axial and shear stresses as seen in Figs. 4-6, where shear creep deformations are influenced by the superimposed axial stresses (including hydrostatic pressure) and axial creep deformations are affected by the superimposed shear stresses.

It is to be noted that two different set of curves are predicted by each model for the two stress states as shown in Figs. 4 and 5 and they have the same trend as the experiments. This indicates that both the integral and the differential models have some predictive capabilities of the hydrostatic stress effect. For the sake of comparison, the predicted creep curves by a linear model, which can be derived by assuming all nonlinear functions in the integral model to be unity, are also shown in Figs. 4 and 6. The linear model predicts a single curve for both cases, which portrays no axial/shear stress interaction.

The prediction capability of the differential model can be improved by incorporating more than two Kelvin elements. This, however, will be subject of a forthcoming paper and will not be discussed further herein.

Non-Proportional Path Tests

a) Strain-controlled tests

The predicted axial and shear stress-strain curves by the nonlinear models for the two non-proportional strain paths at the octahedral shear strain rate of [10.sup.-4] [s.sup.-1] are depicted in Figs. 7 and 8, respectively. The predictions by both models are in qualitative agreement for axial and shear stress responses.

b) Strain-rate effect

It is to be noted that for linear viscoelastic models there is no strain-rate dependency, only time effects. However, the two nonlinear models used herein do predict the strain rate effect, as indicated below. The integral model predictions for the axial and shear stress-strain curves for the corner tests at two octahedral shear strain rates of [10.sup.-4] [s.sup.-1] and 1 [10.sup.-5] [s.sup.-1] are shown in Figs. 10 and 11, respectively. The predictions of the differential model do not differ significantly from those shown in the above Figures; thus, only one model predictions are depicted in the Figures. It is noted that an increased stiffness is predicted with the increase in the octahedral shear strain rate, which qualitatively agrees with the experimental results.

c) Stress-controlled tests

It is seen from Figs. 12 and 13 that the predicted creep deformations for the stress-controlled corner tests are rather poor. The reason for this type of prediction may be attributed to the high equivalent stress values of the tests. For example, the equivalent stress is 33.4 MPa for the stress combination of 21 MPa in axial and 15 MPa in shear. The model parameter fitting procedure described earlier indicates that the highest creep-recovery curve used for the fitting was at a tensile stress value of 23.4 MPa, and at this stress level, the correlation was already poor. Therefore, the prediction for an equivalent stress of 33.4 MPa would be expected to be even worse.

From the comparison between the test results and two chosen typical constitutive models, it seems that the current available models do not appear to be sufficiently accurate to predict the inelastic behavior of the combined axial-shear tests presented herein. Although some qualitative features are captured by both models, the capability for predicting other features such as path-dependency and hydrostatic stress effect must be improved.

CONCLUSIONS

From the results presented in this experimental investigation and the subsequent discussion, the following conclusions are drawn:

* A shear stress with a superimposed normal tensile stress will accelerate and a shear stress with a superimposed normal compressive stress (including a hydrostatic pressure) will retard the shear creep deformation. This behavior could be attributed to the hydrostatic stress influence. A superimposed tensile hydrostatic stress increases the volume, and the molecular arrangement will be less dense, and thus resistance to the inelastic deformation will decrease. Consequently, an increased creep deformation would be expected. A superimposed compressive hydrostatic stress will have a reversed effect, and thus will retard the creep deformation relative to that with a positive volume expansion.

* A superimposed shear stress accelerates the normal creep deformation. This may be attributed to the influence of the second stress invariant. A superimposed shear stress increases the effective (or octahedral shear) stress, and thus the creep deformation will increase under a higher effective (or octahedral shear) stress.

* A normal (shear) stress influences the shear (normal) stress-strain behavior. The presence of a normal (shear) stress reduces the shear (normal) stress that can be sustained, or the stiffness decreases because of the presence of a second stress component.

* A variation within an order of magnitude in the strain rate has no qualitative effect on the interactive behavior of the tested epoxy resin under combined normal and shear stresses. However, the response of the epoxy resin is strain-rate sensitive, with the stress level at the lower strain rate being lower than that at the higher strain rate.

* Normal-shear interaction is also noted from the creep deformation response following different non-proportional stress paths.

* Two currently available constitutive models do predict some normal-shear interactions; however, the accuracy of the predicted inelastic behavior of epoxy resin under a combined normal-shear stress (strain) state is in need of improvement.

ACKNOWLEDGMENTS

The work presented here is part of a general investigation of the behavior of polymeric composites under complex loading and adverse environment. This research is supported by the NOVA/NSERC Senior Industrial Research Chair Program (F. E.) and NSERC research grants to F. E. and Z. X. The authors are also indebted to Dr. J. Wolodko for his assistance in the software modification and valuable suggestions.

REFERENCES

(1.) P. D. Ewing, S. Turner, and J. G. Williams, J. Strain Anal., 8, 83 (1973).

(2.) G. B. McKenna and L. J. Zapas, J. Rheol, 23, 151 (1979).

(3.) R. S. Duran and G. B. McKenna, J. Rheol, 34, 813 (1990).

(4.) H. Lu and W. G. Knauss, Mech. Time-Dependent Mater., 2, 307 (1999).

(5.) M. Kitagawa and H. Takagi, J. Polym. Sci. Part B Polym. Phys., 28, 1943 (1990).

(6.) E. Krempl and C. M. Bordonaro, Int. J. Plast., 14, 245 (1998).

(7.) F. Ellyin and J. D. Wolodko, Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, p. 7, S. Kalluri and P. J. Bonacuse, eds. (1997).

(8.) Y. Hu, Z. Xia, and F. Ellyin, Polym. Polym. Compos., 8, 11 (2000).

(9.) W. G. Knauss and I. Emri, Polym. Eng. Sci., 27, 86 (1987).

(10.) Y. C. Lou and R. A. Schapery, J. Compos. Mater., 5, 208 (1971).

(11.) Y. Hu, Z. Xia, and F. Ellyin, Polym. Polym. Compos., 8, 157 (2000).

(12.) C. F. Popelar and K. M. Liechti, Trans. ASME J. Eng. Mater. Technol., 119, 205 (1997).

(13.) L. Zhang, L. J. Ernst, and H. R. Brouwer, Mech. Mater., 26, 167 (1997).

(14.) K. D. Pae and S. K. Bhateja, J. Macromol. Sci. Rev. Macromol. Chem. Phys., C13, 1 (1975).

(15.) J. Lai and A. Bakker, Comput. Mech., 18, 182 (1996).

(16.) Z. Xia and F. Ellyin, Polym. Polym. Compos., 6, 75 (1998).

(17.) F. Ellyin, Y. Hu, and Z. Xia, Recent Trends in Constitutive Modeling of Advanced Materials, ASME-AMD-Vol. 239. p. 13, M. K. Khraisheh, T. A. Godfrey, and K. J. William, eds. (2000).

(18.) R A. Schapery, Polym. Eng. Sci., 9, 295 (1969).

(19.) D. Peretz and Y. Weitsman, J. Rheol., 27, 295 (1983).

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Table 2. Nonlinear Viscoelasticity Parameters.

[J.sub.0] = 9.957 X [10.sup.-4][(MPa).sup.-1], [DELTA]J = 1.993 X [10.sup.-5][[psi].sup.0.373][(MPa).sup.-1]

[B.sub.0] = 2.134 X [10.sup.-4][(MPa).sup.-1], [DELTA]B = 4.271 X [10.sup.-6][[psi].sup.0.373][(MPa).sup.-1]

[g.sub.0] = 1

[g.sub.1] = {1 for [[sigma].sub.eq] [less than or equal to] 11.293 MPa

[0.505 + 0.0438[[sigma].sub.eq] for [[sigma].sub.eq] > 11.293 MPa

[g.sub.2] = 2.066 + [3.066e.sup.[[sigma].sub.eq]/85.5]

[a.sub.[sigma]] = {1 for [[sigma].sub.kk] [less than or equal to] 12.74 MPa

{1.611 - 0.048[[sigma].sub.kk] for [[sigma].sub.kk] > 12.74 MPa

where [[sigma].sub.eq] = [square root of (3/2 [S.sub.ij][S.sub.ij])], the von Mises equivalent stress [[sigma].sub.kk], the first invariant of a stress tensor

Constitutive modeling of the deformation behavior of solid polymers has attracted considerable interest recently, due to the use of polymeric materials in primary structures. To ensure reliable and safe performance of these structures, stress analysis must be performed, which requires that accurate constitutive equations be available. However, such is not the case.

The choice or development of an appropriate model to predict the mechanical behavior usually necessitates a general understanding of the properties of the material involved, and these properties are mostly found through experiments. The low modulus and sensitivity to the environment of polymeric materials make the experimental investigation of polymers a fairly challenging task, especially when multiaxial loading is applied.

In plasticity, axial-torsional tests are often used to study the interaction between normal and shear stress (strain) components. These tests are employed to discriminate among different constitutive models of plastic deformation. An appropriate constitutive model for polymers should also be able to describe the mechanical response of axial-torsional tests. In polymeric composite structures where polymers are used as matrix materials, the stress state in the matrix is generally of a multiaxial nature. Therefore, the interaction between different stress (strain) components is frequently encountered in these structures. Research on interaction of normal-shear stress (strain) components is an important first step toward modeling more complex stress/strain interaction problems encountered in fiber-reinforced composites.

There are a few experimental investigations reported in the literature for polymers subjected to combined normal and shear stresses. A sample of representative studies is described below. Ewing et al. (1) investigated the combined tension-torsion creep behavior of polyethylene with emphasis on the effect of different loading paths on creep response. The interaction between torsion and axial force or volume dilatation was studied by McKenna and co-workers (2. 3). It was found that the torsion influences the axial force and the volume change of polymers like polymethyl methacrylate (PMMA) (2) and epoxy glass (3). Lu and Knauss (4) investigated the nonlinear thermo-mechanical behavior of PMMA under combined axial (tension, compression) and shear (torsional) stress states. It was shown that dilatation affects the shear creep behavior at moderate strains. The above tests, among others, mainly focused on the creep or stress relaxation behavior of polymers. The transient behavior of polyethylene under different stra in histories was investigated by Kitagawa and Takagi (5). Krempl and Bordonaro (6) performed non-proportional loading tests on nylon 66 at room temperature. To the best of our knowledge, no tests have been performed to study the axial-torsional interaction behavior of epoxy resins in which the test temperature is far below the glass transition temperature.

Epoxy resins have certain properties that are different from those of thermoplastics and other unsaturated thermosets. Notable among these is much lower strains to failure. Epoxy resins are widely used in fiber-reinforced composites as matrix materials, and consequently, a thorough understanding of their nonlinear behavior is a prerequisite for the accurate prediction of the mechanical response of epoxy-based composites and design of such composite structures.

This investigation focuses on the axial-torsional interaction effect of an epoxy resin. Thin-walled tubular specimens are subjected to combined constant normal (or shear) stress and varying shear (or normal) stress to study the creep behavior. Different strain paths at constant octahedral shear strain-rate are also applied to the tubular specimens to investigate the transient stress responses. In addition, different constant octahedral shear strain-rates are used to study the rate effects. Furthermore, the normal-shear stress interaction is investigated by creep tests with non-proportional stress histories. It is shown that superimposed normal stress influences the shear strain response and vice versa. Two typical viscoelastic constitutive models are chosen to check their capability to predict the above observed experimental characteristics of the epoxy resin.

SPECIMEN PREPARATION AND TEST FACILITY

Specimen Fabrication

The material used in the experimental investigation was Epon 828, an epoxy resin from the reaction product of epichiorohydrin and bisphenol A, commercially known as ColdCure epoxy. The hardener is a methylenebiscyclohexylamine. The ColdCure epoxy resin and hardener were mixed with 2:1 (volume) ratio and stirred thoroughly, and then poured into a tubular plastic mold. The samples were cured for 1 hour at 65[degrees]C and cooled to room temperature in the oven. Following this, they were held at the ambient temperature for 7 days to reach 100% polymerization prior to machining. Test specimens were manufactured from the cured samples by a computer numerical controlled (CNC) lathe that provides for precise machining of any transition profile geometry. Aluminum end tabs were glued to the tube ends (3-M Type DP-460 Epoxy Adhesive) so that the specimen can be inserted into the gripping system of the test machine. In this way one avoids damaging the specimen extremities during the gripping process. The aluminum end ca ps are also needed because the contact force between the gripping system and tube specimen will relax with time, if the epoxy tube ends are gripped directly. This will then cause the specimen to slip out of the gripping system during tensile tests, and for the oil to leak during differential pressurization (biaxial tests). The final specimen configuration is shown in Fig. 1.

Test Facility

All tests were performed in a servo-hydraulic, computer-controlled, multiaxial testing machine described in ref. (7). This test facility can apply a combination of axial load, differential pressure (inside and outside pressures) and torsion, and is capable of generating all principal stress and strain ratios, as well as change in the direction of the applied principal stresses and strains. For the current investigation, the axial and torsional loading systems were used, except for a shear creep in which a superimposed hydrostatic pressure was applied to the specimen.

A strain gauge in the axial direction served as an axial strain transducer, while a pair of strain gauges mounted at [+ or -] 45[degrees] to the axial direction served as a shear strain transducer. Another pair, mounted on the opposite surface of the tubular specimen (Fig. 2), was used as a supplement to record strains in the [+ or -] 45[degrees] directions. The recorded data from the latter strain gauges served two main purposes. The first was to record the tensile and compressive strains under a pure shear stress. The epoxy resin has been shown to be sensitive to hydrostatic stress, e.g. see ref. (8), these results, therefore, can be used as a base data to compare the effect of a superimposed hydrostatic stress. The second purpose was to confirm that the specimen was aligned coaxially with the grip system. The strain ranges for the axial and shear strain gauges were all calibrated to [+ or -] 5% strain with a resolution of 0.0005%.

The axial and shear stresses imposed on a thin-walled tube subject to an axial load, F, and a torque, T, can be calculated from:

[[sigma].sub.a] = 4F/[pi]([D.sup.2]-[d.sup.2]) (1)

[tau] = 16DT/[pi]([D.sup.4]-[d.sup.4]) (2)

where D and d are the outer and inner diameters of the tube, and [[sigma].sub.a] and [tau] and are the axial and shear stresses, respectively.

The shear stress, [tau], is the stress value on the outer surface of the tube. It is calculated based on the assumption of linear strain distribution across the wall thickness. By using a linear stress variation, the deviation from the average value was [+ or -] 3.96% for the current specimen geometry.

The octahedral shear stress, [[tau].sub.oct], and hydrostatic stress, [[sigma].sub.h], are defined as:

[[tau].sub.oct] = [square root of (2)]/3 [square root of ([[sigma].sup.2.sub.a] + 3[[tau].sup.2])] (3)

[[sigma].sub.h] = [[sigma].sub.a]/3 (without superimposed hydrostatic pressure p) (4a)

or

[[sigma].sub.h] = [[sigma].sub.a]/3 - p

(under superimposed hydrostatic pressure p) (4b)

TEST PROCEDURE

Creep Tests

a) Axial stress effect on shear creep

Two different shear creep tests were conducted (Series I, see Table 1): one with a superimposed constant axial tensile stress (Case 1) and another with a superimposed constant axial compressive stress (Case 2). Both tests have the same axial stress value of 14 MPa and shear stress value of 10 MPa, but the sign of the axial stress varies. These tests were used to investigate the influence of the normal stress on the shear creep behavior.

b) Hydrostatic pressure effect on shear creep

This series of tests included two types of shear creep tests. Here a uniform hydrostatic pressure was superimposed on the shear stress, rather than the axial tensile and compressive stresses as used in the previous test series. The applied shear stress and hydrostatic pressure were 15MPa and OMPa (no hydrostatic pressure) for Case 1, see Table 1, and 15MPa and 17MPa, respectively, for another test (Series II, Case 2). These tests will indicate the effect of the hydrostatic pressure on the shear creep behavior.

c) Shear stress effect on axial creep

The third series of creep tests consisted of an axial tensile test with a superimposed shear stress (Series III in Table 1). In these tests the axial stress value was 14 MPa but the magnitude of the shear stress differed. In this manner the influence of the shear stress on the normal creep behavior can be investigated.

The stresses indicated in Table 1 (except the hydrostatic pressure) were applied within 10 seconds and subsequently held constant for a period of one hour. The resulting creep curves were analyzed for periods beyond 20 seconds, to allow transient loading effects to diminish. For the test with a superimposed hydrostatic pressure, the pressure was applied prior to starting the shear creep test.

Non-Proportional Path

a) Strain-controlled tests

A non-proportional strain-controlled loading scheme is shown in Fig. 3. Two different paths terminate at the same point C in the [[epsilon].sub.a] vs. [gamma] strain space. The testing procedure consisted of loading to a certain axial (shear) strain value that was then held constant. Subsequently, the shear (axial) strain was increased to the coordinate of point C. This is the so-called corner test which, as pointed out by Krempl and Bordonaro (6), is the test often used to discriminate plastic deformation behavior.

All strain paths were loaded at the same strain rate. The rate is defined by the octahedral shear strain rate, [[epsilon].sub.oct] = [square root of (2/9 [(1 + v).sup.2] [[epsilon].sup.2.sub.a] + 1/6 [[gamma].sup.2])], where [[epsilon].sub.a] and [gamma] are axial and shear strains, respectively, a superposed dot designates the time derivative, and v is the Poisson's ratio. The corresponding octahedral shear strain is defined by [[epsilon].sub.oct] = [square root of (2/9 [(1 + v).sup.2] [[epsilon].sup.2.sub.a] + 1/6 [[gamma].sup.2])]. For the corner path test, an octahedral shear strain rate of [10.sup.-4] [s.sup.-1] was chosen, which converts to an axial strain rate of 1.52 X [10.sup.-4] [s.sup.-1] and a shear strain rate of 1.22 x [10.sup.-4] [s.sup.-1]. In Fig. 3, the strain components and the octahedral shear strains, in parentheses, are indicated at points A, B and C. In the calculation of the corresponding axial and shear strains and the strain rates, a value of 0.4 for the Poisson's ratio was used.

b) Strain-rate effect

The effect of strain rate was studied by performing the above tests at a different octahedral shear strain rate of [10.sup.-5] [s.sup.-1]. The corresponding axial and shear strain rates were 1.52 X [10.sup.-5] [s.sup.-1] and 1.22 X [10.sup.-5] [s.sup.-1], respectively, for the corner path tests.

c) Stress-controlled tests

Another set of tests was performed to gain a further insight into the axial-torsional interaction. In these experiments the creep behavior of the epoxy resin was investigated when subjected to different non-proportional stress histories. The loading path was the same type of corner test shown in Fig. 3, but the tests were conducted under a stress control mode, instead of the strain control as depicted in Fig. 3. After reaching point C by different stress paths, the axial and shear stresses were kept constant for an hour. The corresponding axial and shear creep deformations were recorded.

The stress rate for the two stress paths was the same. An effective stress rate defined by [[sigma].sub.eff] = [square root of ([[sigma].sup.2.sub.a] + 3[[tau].sup.2])] was used, where [[sigma].sub.a] and [tau] are axial and shear stress rates, respectively. The effective stress rate was chosen to be 0.1 MPa/s, which corresponds to an axial stress rate of 0.1 MPa/s and a shear stress rate of 0.058 MPa/s. The stress values reached at point C were a combination of 21MPa in axial and 15MPa in shear.

All tests were performed at ambient environments (25[degrees]C and 50% humidity). In order to minimize specimen-to-specimen variations, a single specimen was used as much as possible. This required that the specimen deformation from previous tests be recovered. It was found that as long as the applied stress level was not too high and enough time was allowed for the recovery process, the specimen did indeed recover completely. Confirmatory tests were conducted to verify the observed trends and response values.

RESULTS

Creep Tests

a) Axial stress effect an shear creep

Shear creep responses at two different combinations of axial and shear stresses (see Table 1, series I) are shown in Fig. 4. This figure illustrates the influence of the axial (tensile or compressive) stress on the shear creep behavior of the epoxy resin. It is clear that superimposed axial tension and compression have a readily discernible and different effect on the creep behavior in shear. A shear stress with a superimposed axial tensile stress, case 1, exhibits higher creep deformation than that with the same shear stress but a superimposed axial compressive stress, case 2.

The above results can be explained in terms of the hydrostatic stress states. In the case of a shear stress with a superimposed axial tensile stress, the hydrostatic stress is positive. When the axial stress is compressive, the hydrostatic stress is negative. Therefore, a superimposed positive hydrostatic stress accelerates the shear creep deformation, whereas a superimposed negative hydrostatic stress retards the shear creep deformation.

b) Hydrostatic pressure effect on shear creep

As shown in Fig. 4. a shear stress with a superimposed axial tensile stress, case 1. exhibits a higher creep deformation than that with the same shear stress but a superimposed axial compressive stress, case 2, and this difference can be attributed to the hydrostatic stress effect. However, in the above tests, the hydrostatic stress is only one-third of the axial stress; therefore, the hydrostatic stress values are relatively small and, thus, the difference in the creep strain response is not significant. To further explore the hydrostatic stress effect, two shear creep tests, one with a superimposed hydrostatic pressure and another one without it, were conducted and the shear strain responses are shown in Fig. 5. Here, the difference is quite noticeable, i.e., a shear stress with a superimposed hydrostatic pressure retards the creep deformation.

c) Shear stress effect on axial creep

Figure 6 depicts the influence of a superimposed shear stress on the axial tensile creep behavior. It can be seen that a superimposed shear stress has a significant effect on the axial creep response. A superimposed shear stress accelerates the creep deformation. This can also be attributed to the increase in the second stress invariant. A superimposed shear stress increases the effective (or octahedral shear) stress, and the creep deformation accelerates under a higher effective (or octahedral shear) stress.

Non-Proportional Path Tests

a) Strain-controlled tests

The axial stress responses for the two loading paths OAC and OBC at an octahedral shear strain rate of [10.sup.-4] [s.sup.-1] are shown in Fig. 7. The solid square symbols represent the stress response for the loading path OAC and the hollow ones for the loading path OBC. For both strain paths, axial stress increases with the increase of strain until a strain value of 0.742% is reached. Prior to reaching the aforementioned axial strain, the axial stress response for path OBC is slightly below that of path OAC. This difference can only be attributed to the presence of a prior shear stress for the path OBC. Therefore, this figure shows that the presence of a prior shear stress causes a reduction in the stiffness of the axial stress-strain behavior.

The shear response for the same strain paths is depicted in Fig. 8. Again two distinct curves are observed, but the trend is reversed, i.e., the shear stress-strain curve for the path OBC is slight higher than that of OAC because of the presence of a prior axial stress in the latter path. Therefore, the existence of a prior axial stress also causes a reduction in the stiffness of the shear stress-strain curve.

With reference to Fig. 7 and path OAC, the axial stress drops from point A to the lowest point C by about 3.06 MPa when the shear stress, path AC, is applied. The corresponding drop in the shear stress from point B to lowest point C in Fig. 8 is about 1.32 MPa. In Figs. 7 and 8 there are two points labeled C. They are referred to the axial (shear) stresses reached by the different strain paths. The axial and shear stress drops observed in Figs. 7 and 8 are due to the stress relaxation while the axial or shear strains are kept constant. There may also be an interaction influence. In order to determine the contribution to the total stress drop by each of the above factors, another test was performed in which the axial strain was first applied at a constant octahedral shear strain rate of [10.sup.-4] [s.sup.-1]. After it reached point A, the strain was kept constant instead of applying a shear strain as in path OAC of Fig. 3. The corresponding stress response (OAC') is shown in Fig. 9, in conjunction with that o f path OAC. Note that the stress drop from A to C' is due to stress relaxation only, while the stress drop from A to C is a result of both the stress relaxation and shear stress interaction. It is seen from the figure that about 1/3 of the total stress drop can be attributed to the shear stress interaction effect.

b) Strain-rate effect

Figures 10 and 11 show the axial stress versus axial strain and the shear stress versus shear strain for two octahedral shear strain rates of [10.sup.-4] [s.sup.-1] and [10.sup.-5] [s.sup.-1]. It is noted that there is no significant relative difference between the stress responses due to the path effect as a result of change in the strain rate of one order of magnitude. However, the change in the strain rate has a noticeable influence on the stiffness of the stress-strain response, i.e., the elastic modulus is a function of time.

c) Stress-controlled tests

The axial and shear strain responses for a corner test with a maximum axial stress of 21 MPa and a maximum shear stress of 15 MPa, are shown in Figs. 12 and 13, respectively. The figures show different strain responses for the two loading paths, with the creep deformation from path OBO being higher than that from path OAC for both the axial and shear strain components.

The experimental results discussed herein clearly show that the stress (strain) response of the epoxy resin is stress or strain component dependent, that is, an axial stress (strain) will influence the shear strain (stress) response, and vice versa. The influence is observed in both the transient and creep responses.

CONSTITUTIVE MODELING

A number of constitutive models have been proposed to predict the nonlinear viscoelastic response. For example, the free volume theory of Knauss and Emri (9) and Lou and Schapery thermodynamic based model (10) are among those well-known and often quoted. This type of integral representation is convenient for the stress and strain analysis, and the experimental data required for the determination of material constants is not onerous. However, their application in the multiaxial stress states has not been fully explored (11).

There is no interaction between the axial stress (strain) and the shear stress (strain) in the linear range of a viscoelastic material; therefore, axial-shear interaction is a nonlinear phenomenon. In certain cases, nonlinear viscoelastic models are obtained by modifying linear viscoelastic models by replacing ordinary tune in the kernel functions with a strain (stress) modified time. Collectively these models are known as "strain clock" and "stress clock" models depending upon whether the modified time is made to depend on the strain or stress, respectively. The free volume model developed by Knauss and Emri (9) is a case in point, in which the strain clock depends upon the first strain invariant, and this dependence is motivated by the free volume consideration. As pointed out by Krempl and Bordonaro (6), such a model would not be able to reproduce the results reported here because a prior torsion has no influence on a subsequent tension loading in this model.

A simple way to predict coupling between normal and shear stresses of isotropic materials in small deformation would be to include a second invariant although this is not necessarily the only form. An attempt was made by Popelar and Liechti (12) to formulate such a model. Although originally it was proposed to consider the nonlinearity for shear-dominated loadings, it may also be able to predict the axial-shear interaction because of the inclusion of the effective strain in the reduced time (intrinsic time). However, little experimental verification has been carried out on this model, and its predictive capabilities must be further explored.

The Schapery model (10) can, in principle, account for the torsion-axial coupling because the equivalent stress can be included in the nonlinear functions, as demonstrated by Zhang et al. (13). In their paper, a set of combined torsion-axial creep tests was performed, and a good agreement between test results and model predictions was obtained. The equivalent stress, [[sigma].sub.eq], initially defined in the above constitutive model accounted for the distortion energy only, i.e., it was the von Mises equivalent stress. However, as shown in our creep tests and others, see e.g. (4, 14), hydrostatic stress has a discernable effect on the shear creep response of polymers. The hydrostatic stress is related to the bulk dilatation energy. If only an equivalent stress based on the von Mises is used (distortion energy), then the hydrostatic stress influence is not accounted for. Therefore, the equivalent stress, [[sigma].sub.eq], has to be modified for polymers. Lai and Bakker (15) have attempted to include the hydr ostatic stress influence in a modified Schapery model. However, it has been shown (11) and will be shown later that the model does not adequately simulate the hydrostatic stress influence on the mechanical behavior of epoxy resins.

To illustrate the predictive capability of the current constitutive models, two typical nonlinear viscoelastic models, one by Lai and Bakker (15), a modified Schapery model in the integral form, and the other by Xia and Ellyin (16), a differential form model, are chosen to predict the mechanical response of the epoxy resin under the aforementioned loading paths. The equations for the two nonlinear viscoelastic constitutive models are summarized below.

Nonlinear Constitutive Models

a) Integral representation

For a thermo-rheologically simple isotropic viscoelastic material under a multiaxial state of stress, the constitutive law can be expressed as:

[[epsilon].sub.ij] = 1/2 [J.sub.0][g.sub.0][s.sub.ij](t) + 1/2 [g.sub.1] [[integral].sup.t.sub.0] [DELTA]J[[psi](t) - [psi]'([tau])] d([g.sub.2][s.sub.ij])/d[tau] d[tau] + 1/9 [B.sub.0][g.sub.0][[delta].sub.ij][[sigma].sub.kk](t) + 1/9 [[delta].sub.ij][g.sub.1] [[integral].sup.t.sub.0] [DELTA]B[[psi](t) - [psi]'([tau])] d([g.sub.2][[sigma].sub.kk])/d[tau] d[tau] (5)

where [J.sub.0] and [B.sub.0] are instantaneous shear and bulk compliances; [DELTA]J and [DELTA]B are transient shear and bulk compliances, respectively; [psi] and [psi]' are reduced times,

[psi] = [[integral].sup.t.sub.0] d[tau]/[a.sub.[sigma]] and [psi]' = [[integral].sup.[tau].sub.0] d[tau]/[a.sub.[sigma]];and [g.sub.0] and [a.sub.[sigma]] are nonlinear functions of [[sigma].sub.kk], the first invariant of a stress tensor; [g.sub.1] and [g.sub.2] are nonlinear functions of the von Mises equivalent stress.

b) Differential representation

In this model, the total strain rate {[[epsilon].sub.t]} is assumed to be the sum of the elastic and creep strain rates, {[[epsilon].sub.e]} and {[[epsilon].sub.c]}, respectively, i.e.,

{[[epsilon].sub.t]} = {[[epsilon].sub.e]} + {[[epsilon].sub.c]} (6)

The elastic strain rate is calculated through Hooke's law,

{[sigma]} = E[[A].sup.-1]{[[epsilon].sub.e]} (7)

For a number of Kelvin (Voigt) elements connected in series, creep strain rate {[[epsilon].sub.c]} is the sum of the strain rate of each element {[[epsilon].sub.ci]}, which is nonlinearly related to applied stress {[sigma]},

{[[epsilon].sub.c]} = [summation over (n/i=1)]{[[epsilon].sub.ci]} = [summation over (n/i=1)]([a.sub.i][A][[sigma].sup.[[alpha].sub.i]-1.sub.eqM]{[sigma]} - [b.sub.i]{[[epsilon].sub.ci]}) (8)

where [a.sub.i], [b.sub.i] and [[alpha].sub.i] (i = 1, 2 ..., n) are material constants, E is Young's modulus, [A] is a matrix related to the Poisson's ratio, [[sigma].sub.eqM] is a modified von Mises equivalent stress, which includes the hydrostatic stress term (17).

All the model constants and functions for the integral representation are calibrated from a set of uniaxial creep-recovery curves at different stress levels (11). The stress in creep tests varied from 2.8 MPa to 23.4 MPa. The adopted method to determine functional forms and constants was that advocated by Schapery (18) and widely used by other researchers, see e.g., Peretz and Weitsman (19). The basic characterization procedure involves fitting the uniaxial form of Eq 5 to linear creep-recovery curve at low stress level, as a first step, to determine [J.sub.0], [B.sub.0], [DELTA]J and [DELTA]B. Then the uniaxial integral equation is fitted with each creep-recovery curve and the stress dependence of [g.sub.0], [g.sub.1], [g.sub.2] and [a.sub.[sigma]] is determined. The function relations between [g.sub.0], [g.sub.1], [g.sub.2], [a.sub.[sigma]] and the applied stresses are finally fitted by least square method. These model constants and functions for the integral representation are listed in Table 2.

In order to compare the prediction capability of these two models, the same creep-recovery curves were used to obtain the material constants for the differential model. The uniaxial form of Eq 8 indicates that the strain recovery depends only on the constants [b.sub.1] and [b.sub.2] ([sigma] = 0), therefore the two material constants can be determined first from the strain recovery curve at a given stress level (16). Although any strain recovery curve can be used for the calibration of [b.sub.1] and [b.sub.2], the average values of [b.sub.1] and [b.sub.2] from all stress levels were used here. Subsequently, the constants E, [a.sub.1], [a.sub.2], [[alpha].sub.1] and [[alpha].sub.2] are determined based on the best fit to the creep curves as follow:

For a constant stress [[sigma].sub.0], the uniaxial creep strain can be reformulated from the above differential relations as

[epsilon] = [[sigma].sub.0]/E + [[sigma].sup.[[alpha].sub.1].sub.0]/[E.sub.1] (1 - [e.sup.-[b.sub.1]t]) + [[sigma].sup.[[alpha].sub.2].sub.0]/[E.sub.2] (1 - [e.sup.-[b.sub.2]t]) (9)

where [E.sub.i] = [b.sub.i]/[a.sub.i] (i = 1, 2). Fitting the above equation with the experimental creep curve at the stress level [[sigma].sub.0], the values for E, [[sigma].sup.[[alpha].sub.1].sub.0]/[E.sub.1] and [[alpha].sup.[[alpha].sub.2].sub.0]/[E.sub.2] can be obtained. When fitting Eq 9 with all the creep-recovery curves at different stress levels, two curves, [[sigma].sup.[[alpha].sub.1]]/[E.sub.1] ~ [sigma] and [[sigma].sup.[[alpha].sub.2]]/[E.sub.2] ~ [sigma], are obtained. Therefore, [[alpha].sub.1], [E.sub.1], [[alpha].sub.2] and [E.sub.2] can be determined by a least-square fitting of these two curves. Thus, the material constants for the differential representation are found to be (11):

[a.sub.1] = 3.4 X [10.sup.-8] [(MPa).sup.-1.47], [[alpha].sub.1] = 1.47, [b.sub.1] = 0.001,

[a.sub.2] = 2.70 X [10.sup.-12] [(MPa).sup.-4.38],

[[alpha].sub.2] = 4.38, [b.sub.2] = 1 X [10.sup.-4],

E = 2450 MPa and v = 0.4

which represents two Kelvin elements and one elastic element in series in a uniaxial stress representation.

COMPARISON OF MODEL PREDICTIONS WITH EXPERIMENTAL DATA

Creep Teats

The predicted shear and axial creep deformations by the two models are shown in Figs. 4-6. It is noted that the predictions for the uniaxial tensile stress state (series III, case 2) are in good agreement with the experiment data, Fig. 6. This is to be expected since the parameters for the two viscoelastic constitutive models were fitted to the uniaxial tensile creep curves. However, for all other stress states, the predicted creep curves generally overestimate the experimental results. The predictions are in qualitative agreement with the experimental results, noting that the two models do predict some interaction between axial and shear stresses as seen in Figs. 4-6, where shear creep deformations are influenced by the superimposed axial stresses (including hydrostatic pressure) and axial creep deformations are affected by the superimposed shear stresses.

It is to be noted that two different set of curves are predicted by each model for the two stress states as shown in Figs. 4 and 5 and they have the same trend as the experiments. This indicates that both the integral and the differential models have some predictive capabilities of the hydrostatic stress effect. For the sake of comparison, the predicted creep curves by a linear model, which can be derived by assuming all nonlinear functions in the integral model to be unity, are also shown in Figs. 4 and 6. The linear model predicts a single curve for both cases, which portrays no axial/shear stress interaction.

The prediction capability of the differential model can be improved by incorporating more than two Kelvin elements. This, however, will be subject of a forthcoming paper and will not be discussed further herein.

Non-Proportional Path Tests

a) Strain-controlled tests

The predicted axial and shear stress-strain curves by the nonlinear models for the two non-proportional strain paths at the octahedral shear strain rate of [10.sup.-4] [s.sup.-1] are depicted in Figs. 7 and 8, respectively. The predictions by both models are in qualitative agreement for axial and shear stress responses.

b) Strain-rate effect

It is to be noted that for linear viscoelastic models there is no strain-rate dependency, only time effects. However, the two nonlinear models used herein do predict the strain rate effect, as indicated below. The integral model predictions for the axial and shear stress-strain curves for the corner tests at two octahedral shear strain rates of [10.sup.-4] [s.sup.-1] and 1 [10.sup.-5] [s.sup.-1] are shown in Figs. 10 and 11, respectively. The predictions of the differential model do not differ significantly from those shown in the above Figures; thus, only one model predictions are depicted in the Figures. It is noted that an increased stiffness is predicted with the increase in the octahedral shear strain rate, which qualitatively agrees with the experimental results.

c) Stress-controlled tests

It is seen from Figs. 12 and 13 that the predicted creep deformations for the stress-controlled corner tests are rather poor. The reason for this type of prediction may be attributed to the high equivalent stress values of the tests. For example, the equivalent stress is 33.4 MPa for the stress combination of 21 MPa in axial and 15 MPa in shear. The model parameter fitting procedure described earlier indicates that the highest creep-recovery curve used for the fitting was at a tensile stress value of 23.4 MPa, and at this stress level, the correlation was already poor. Therefore, the prediction for an equivalent stress of 33.4 MPa would be expected to be even worse.

From the comparison between the test results and two chosen typical constitutive models, it seems that the current available models do not appear to be sufficiently accurate to predict the inelastic behavior of the combined axial-shear tests presented herein. Although some qualitative features are captured by both models, the capability for predicting other features such as path-dependency and hydrostatic stress effect must be improved.

CONCLUSIONS

From the results presented in this experimental investigation and the subsequent discussion, the following conclusions are drawn:

* A shear stress with a superimposed normal tensile stress will accelerate and a shear stress with a superimposed normal compressive stress (including a hydrostatic pressure) will retard the shear creep deformation. This behavior could be attributed to the hydrostatic stress influence. A superimposed tensile hydrostatic stress increases the volume, and the molecular arrangement will be less dense, and thus resistance to the inelastic deformation will decrease. Consequently, an increased creep deformation would be expected. A superimposed compressive hydrostatic stress will have a reversed effect, and thus will retard the creep deformation relative to that with a positive volume expansion.

* A superimposed shear stress accelerates the normal creep deformation. This may be attributed to the influence of the second stress invariant. A superimposed shear stress increases the effective (or octahedral shear) stress, and thus the creep deformation will increase under a higher effective (or octahedral shear) stress.

* A normal (shear) stress influences the shear (normal) stress-strain behavior. The presence of a normal (shear) stress reduces the shear (normal) stress that can be sustained, or the stiffness decreases because of the presence of a second stress component.

* A variation within an order of magnitude in the strain rate has no qualitative effect on the interactive behavior of the tested epoxy resin under combined normal and shear stresses. However, the response of the epoxy resin is strain-rate sensitive, with the stress level at the lower strain rate being lower than that at the higher strain rate.

* Normal-shear interaction is also noted from the creep deformation response following different non-proportional stress paths.

* Two currently available constitutive models do predict some normal-shear interactions; however, the accuracy of the predicted inelastic behavior of epoxy resin under a combined normal-shear stress (strain) state is in need of improvement.

ACKNOWLEDGMENTS

The work presented here is part of a general investigation of the behavior of polymeric composites under complex loading and adverse environment. This research is supported by the NOVA/NSERC Senior Industrial Research Chair Program (F. E.) and NSERC research grants to F. E. and Z. X. The authors are also indebted to Dr. J. Wolodko for his assistance in the software modification and valuable suggestions.

REFERENCES

(1.) P. D. Ewing, S. Turner, and J. G. Williams, J. Strain Anal., 8, 83 (1973).

(2.) G. B. McKenna and L. J. Zapas, J. Rheol, 23, 151 (1979).

(3.) R. S. Duran and G. B. McKenna, J. Rheol, 34, 813 (1990).

(4.) H. Lu and W. G. Knauss, Mech. Time-Dependent Mater., 2, 307 (1999).

(5.) M. Kitagawa and H. Takagi, J. Polym. Sci. Part B Polym. Phys., 28, 1943 (1990).

(6.) E. Krempl and C. M. Bordonaro, Int. J. Plast., 14, 245 (1998).

(7.) F. Ellyin and J. D. Wolodko, Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, p. 7, S. Kalluri and P. J. Bonacuse, eds. (1997).

(8.) Y. Hu, Z. Xia, and F. Ellyin, Polym. Polym. Compos., 8, 11 (2000).

(9.) W. G. Knauss and I. Emri, Polym. Eng. Sci., 27, 86 (1987).

(10.) Y. C. Lou and R. A. Schapery, J. Compos. Mater., 5, 208 (1971).

(11.) Y. Hu, Z. Xia, and F. Ellyin, Polym. Polym. Compos., 8, 157 (2000).

(12.) C. F. Popelar and K. M. Liechti, Trans. ASME J. Eng. Mater. Technol., 119, 205 (1997).

(13.) L. Zhang, L. J. Ernst, and H. R. Brouwer, Mech. Mater., 26, 167 (1997).

(14.) K. D. Pae and S. K. Bhateja, J. Macromol. Sci. Rev. Macromol. Chem. Phys., C13, 1 (1975).

(15.) J. Lai and A. Bakker, Comput. Mech., 18, 182 (1996).

(16.) Z. Xia and F. Ellyin, Polym. Polym. Compos., 6, 75 (1998).

(17.) F. Ellyin, Y. Hu, and Z. Xia, Recent Trends in Constitutive Modeling of Advanced Materials, ASME-AMD-Vol. 239. p. 13, M. K. Khraisheh, T. A. Godfrey, and K. J. William, eds. (2000).

(18.) R A. Schapery, Polym. Eng. Sci., 9, 295 (1969).

(19.) D. Peretz and Y. Weitsman, J. Rheol., 27, 295 (1983).

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Table 1 The Applied Stress States for Creep Tests. Hydrostatic Shear Stress Axial Stress Pressure Series No. Case No. (MPa) (MPa) (MPa) I 1 10 14 -- 2 10 -14 -- II 1 15 -- 0 2 15 -- 17 III 1 10 14 -- 2 0 14 --

Table 2. Nonlinear Viscoelasticity Parameters.

[J.sub.0] = 9.957 X [10.sup.-4][(MPa).sup.-1], [DELTA]J = 1.993 X [10.sup.-5][[psi].sup.0.373][(MPa).sup.-1]

[B.sub.0] = 2.134 X [10.sup.-4][(MPa).sup.-1], [DELTA]B = 4.271 X [10.sup.-6][[psi].sup.0.373][(MPa).sup.-1]

[g.sub.0] = 1

[g.sub.1] = {1 for [[sigma].sub.eq] [less than or equal to] 11.293 MPa

[0.505 + 0.0438[[sigma].sub.eq] for [[sigma].sub.eq] > 11.293 MPa

[g.sub.2] = 2.066 + [3.066e.sup.[[sigma].sub.eq]/85.5]

[a.sub.[sigma]] = {1 for [[sigma].sub.kk] [less than or equal to] 12.74 MPa

{1.611 - 0.048[[sigma].sub.kk] for [[sigma].sub.kk] > 12.74 MPa

where [[sigma].sub.eq] = [square root of (3/2 [S.sub.ij][S.sub.ij])], the von Mises equivalent stress [[sigma].sub.kk], the first invariant of a stress tensor

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Author: | Hu, Yafei; Ellyin, Fernand; Xia, Zihui |
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Publication: | Polymer Engineering and Science |

Article Type: | Statistical Data Included |

Geographic Code: | 1USA |

Date: | Nov 1, 2001 |

Words: | 6451 |

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