# Viscoelastic relaxations in polyepoxide joints related to the strength of bonded structures at impact rate shear loading.

INTRODUCTION

The impact behavior of new, lightweight bonded structures requires careful examination. Adhesives, being polymeric materials, have properties that are usually rate dependent, because of their viscoelasticity. It is by no means certain that their performance under static conditions will be the same as under dynamic ones. There is, therefore, the need to give designers some confidence that under impact conditions, bonded joints will maintain their structural integrity. Unfortunately, few studies have been carried out to investigate properties of bonded joints at medium and high strain rates. For this purpose, a new experimental setup based on the Charpy technique (the inertia wheel) was used to characterize the impact resistance of structural adhesive joints ([Mathematical Expression Omitted] to [10.sup.4] [s.sup.-1]) under shear loading. The study was completed by means of static tests and temperature tests (-30, 40, 60, 80 [degrees] C). We found there was a critical strain rate [Mathematical Expression Omitted] (T) (which is a function of temperature), above which the joint becomes more resistant.

This change of behavior, which affects not only the yield stress but also the failure energy, was successfully explained using the Bauwens approach (1) which consists of a modification of the Ree-Eyring (2) theory.

In this paper, we show that the Escaig model (3) confirms the role played by the [Beta] mechanical loss peak on the strain rate sensitivity of the yield stress. It was indeed possible to derive the activation energy associated with the [Beta] transition of our modified epoxy resin from both models, which is found to be in good agreement with the values given in the literature. The Escaig model has the advantage of containing two relevant parameters: the volume activation [V.sub.a], which reflects the local flexibility of the macromolecular chains, and a nonlinear parameter [[Tau].sub.i], which determines the effect of changes in the polymer structure. Furthermore, the evolution of both parameters allows us to explain accurately the limitation of these models at low temperatures (-30 [degrees] C).

EXPERIMENTAL PROCEDURES

Materials

The two structural adhesives investigated were industrial one-part adhesives produced by CECA Adhesifs: An unmodified epoxy resin and a modified one. The two components of the unmodified resin are i) the epoxy resin (a blend of different DGEBA by-products) and ii) the curing agent (Dicyandiamine). The modified epoxy resin was obtained by incorporating in the resin a block copolymer modifier (commercially secret) and fillers (aluminium, amorphous silica, and calcium carbonate). The substrate was a galvanized steel sheet (thickness: 1.5 mm). Before bonding, steel sheets were simply cleaned in an ethyl acetate bath.

Sample

The specimens are made of two galvanized steel sheets bonded by epoxy resin [ILLUSTRATION FOR FIGURE 1 OMITTED]. The two steel sheets are clamped together with PTFE spacers at each side of the substrate to establish its nominal thickness (0.5 mm).

The adhesive was cured by heating in an oven, 30 min at 180 [degrees] C. To carry out loading using the impact test system, the sample was fixed onto rigid supports. Fixation was obtained by bonding with a cyanacrylate adhesive applied to the whole surface of the steel area, which is sufficient to strongly attach it to the support [ILLUSTRATION FOR FIGURE 2 OMITTED].

High Strain Rate Tests

The high strain rate tests were carried out on a specific experimental setup, the so-called "Inertia Wheel" [ILLUSTRATION FOR FIGURE 3 OMITTED]. This new device based on the inertia technique is very suitable; the tangential velocity of the wheel can be adjusted very accurately (from 1 m/s to 20 m/s). When the wheel reaches the nominal tangential velocity, the specimen rotates just before the hammer fixed on the wheel by means of a pendulum mechanism activated by a pneumatic jack [ILLUSTRATION FOR FIGURE 4 OMITTED].

A charge amplifier converts the electric charge supplied by the piezoelectric transducer into a voltage signal which is proportional to the applied force. The signal is stored in a numerical oscilloscope. A Zimmer OHG extensometer (bandwidth: 200 kHz) is used to follow the relative displacement of the two substrates by means of two marks stuck on each support [ILLUSTRATION FOR FIGURE 5 OMITTED]. The high energy stored by the wheel compared to the energy required to break a joint allows us to obtain a strain rate that is nearly constant throughout the test [ILLUSTRATION FOR FIGURE 6 OMITTED]. Since the frequency response of the quartz washer is 40 kHz and the smallest time to failure in any test is [approximately]30 [[micro]seconds], this system is obviously adequate (4, 5).

Since the displacements due to the rigid supports are not negligible, they must be considered. For this reason, further tests with the supports alone were carried out and then, the apparatus compliance is taken into account [ILLUSTRATION FOR FIGURE 6 OMITTED].

Figure 6 shows that the polymer does not exhibit a purely linear behavior. The maximum point on the curve is sufficiently remarkable to consider it the yield point to account for the pre-plastic deformation of the epoxy, related to the multiplication of shears bands, which self-accelerate (6). Thus, the yield stress and the critical energy can readily be obtained from the following equations:

[[Tau].sub.y] = [F.sub.max]/[A.sub.joint] (1)

[E.sub.c] = [V.sub.ham]/[A.sub.joint] [[integral of] [F.sub.1](t)dt between limits [t.sub.f] and 0 - [integral of] [F.sub.2](t)dt between limits [t.sub.s] and 0] (2)

Where:

[V.sub.ham]: velocity of the hammer

[V.sub.ham] [multiplied by] [t.sub.f]: displacement of the adhesive joint and the supports.

[V.sub.ham] [multiplied by] [t.sub.s]: displacement due to the only supports.

[A.sub.joint]: area of the joint

Tests at Lower Strain Rates

The shearing tests at lower speeds were carried out using an Instron test machine ([10.sup.-2] to [10.sup.1] [s.sup.-1]) and a hydraulic tensile machine ([10.sup.2] to [10.sup.3] [s.sup.-1]).

RESULTS AND DISCUSSION

We successfully applied the Bauwens approach to describe the behavior of the modified epoxy joint (0.5 mm thick joint) over a wide range of strain rates and temperatures. The important results concerning this approach are summarized in a first step. Then, to confirm the important role played by the viscoelastic properties of the adhesive on the strain rate sensitivity of the yield stress, we extended the study by applying the Escaig model, which has the advantage of taking account of the structural modifications of the polymer. That will be the subject of the last part of this contribution.

The Eyring-Bauwens Model

If the polymer is capable of exhibiting a secondary transition [Beta], (associated to the local relaxation modes of the macromolecular structure or the relaxation of the side groups), Bauwens showed that at high strain rates, the variation of the yield stress with the strain rate can be described by the generalized theory proposed by Ree-Eyring (2):

[Mathematical Expression Omitted] (3)

where T is the absolute temperature and R is the universal gas constant. The parameters [A.sub.[Alpha]], [C.sub.[Alpha]], [A.sub.[Beta]], [C.sub.[Beta]] are characteristics of the material and [Q.sub.[Alpha]], [Q.sub.[Beta]] are the energy barriers of respectively the [Alpha] relaxation and the [Beta] relaxation. These barrier energies have to be overcome so that a segment macromolecular can jump from one equilibrium position to another. This model considers that below a critical strain rate [Mathematical Expression Omitted], only the activation of the main relaxation ([Alpha] relaxation), associated to the relaxation of the crosslinked polymer network, requires a mechanical energy. On the other hand, above [Mathematical Expression Omitted], both [Alpha] and [Beta] need to be surmounted. In epoxy systems, this [Beta] peak is related to motions of the ether linkage segments in the epoxy component of the resin (7):

[Mathematical Expression Omitted]

The secondary transition ([Beta]) of the modified epoxy has been located at -60 [degrees] C, from dynamic mechanical spectroscopy measurements at 1 Hz.

Comparison Between Bauwens Model and Experimental Results

Figure 7 shows the ratio of the yield stress to absolute temperature versus the logarithm of [Mathematical Expression Omitted] at different temperatures, for a 0.5-mm-thick joint. The full lines plotted in Fig. 7 correspond to the theorical variation using the following Eyring equation:

[Mathematical Expression Omitted] (4)

with [X.sub.[Alpha]] = ln(2[C.sub.[Alpha]]) + [Q.sub.[Alpha]]/RT and [Mathematical Expression Omitted]

For each temperature, the parameters of Eq 4 are fitted to the experimental results, allowing us to determine [Q.sub.[Alpha]], [Q.sub.[Beta]], [C.sub.[Alpha]] and [C.sub.[Beta]] (Table 1).

Discussion

It is worth noting that the parameters are constant whatever the temperature (except for -30 [degrees] C). Furthermore, the [Beta] activation energy agrees with that given in the literature for amine-based DGEBA (8) (Table 2). To check the accuracy of this interpretation, a Fourier analysis (9) was also performed to obtain the complete frequency spectrum of the impact pulse corresponding to [Mathematical Expression Omitted]. Every [Mathematical Expression Omitted] is linked to a limit frequency [f.sub.c](T). These values were used in the Arrhenius time-temperature conversion, allowing us to calculate an activation energy [Q.sub.[Beta]] close to those found by other authors or calculated from the Eyring equation, reinforcing the Bauwens approach (Table 2). Therefore, in the temperature range from 24 to 80 [degrees] C, the yield behavior of the modified epoxy joint, above [Mathematical Expression Omitted], may be described by the sum of the partial stresses [[Tau].sub.[Alpha]] and [[Tau].sub.[Beta]] required to free the different kinds of molecular motions implied in the deformation process.

Dynamic mechanical analysis, performed at 1 Hz, revealed the existence of the [Beta] transition for the modified epoxy adhesive at -60 [degrees] C. But let us note that even though the secondary transition is located at a low temperature does not mean that, at room temperature, the local molecular motions are free to move. For instance, an impact stress pulse corresponds to high-frequency measurement, which, by the time-temperature equivalence principle, has the effect of shifting the secondary transition to higher temperatures.

In that way, depending on the temperature and the strain rate, the molecular motions of the [Beta] relaxation [TABULAR DATA FOR TABLE 1 OMITTED] may be frozen or not. The point transition is given by [Mathematical Expression Omitted].

On the other hand, the significant diminution of An and [C.sub.[Beta]] at -30 [degrees] C is likely to arise as a result of microstructural changes in the polymer: at high strain rates and low temperatures, the deformation is confined to a reduced volume, which promotes a concentration of adiabatic shear bands and a decrease of the yield stress (characterized by a decrease of [A.sub.[Beta]]). The deformation is no longer homogeneous (nonlinear behavior), and the parameter [C.sub.[Beta]] (which contains a frequency factor), related to the relaxation time [[Tau].sub.[Beta]], becomes sensitive to temperature (2, 6).

Unfortunately, it is difficult to explain this phenomenon from the Eyring-Bauwens model since it does not account for the structural modifications of the polymer, and heterogeneous fields of deformation.

The Escaig model (3), developed in the eighties, uses a complex formalism, but has the advantage of being based on a nonlinear approach. It is thus much more appropriate for describing the inelastic deformation of polymers. The recent results obtained from this model are reported below.

The Escaig Model

The yielding behavior of thermosets is reviewed in terms of a thermal activated dislocation propagation mechanism. Entanglements are viewed as a local barriers [Delta][H.sub.o] to be overcome with the help of thermal fluctuations and mechanical work.

The jumping of the local barrier, required for the propagation of a dislocation, may be written as follows:

[Delta][H.sub.o] = [Delta][H.sub.a] + ([Tau] - [[Tau].sub.t])[V.sub.a] (5)

Where:

* [V.sub.a] is associated to the spatial extension of the shear bands located where the molecular motions are very important.

* [[Tau].sub.i] is an internal stress that accounts for the entropic effects due to the chain elongation. It is thus directly proportional to [[Tau].sub.[Alpha]].

* [Delta][H.sub.a] is the activation energy, which accounts for the thermal fluctuations.

[V.sub.a] and [Delta][H.sub.a] characterize, respectively, the stress and the temperature sensitivity of the strain rate. These important activation parameters make it possible to obtain useful information about the polymer deformation mechanisms. Indeed, if [V.sub.a] is independent of the stress and the temperature, it is possible to relate the activation volume and the activation enthalpy to parameters measured by experiment (10):

[Mathematical Expression Omitted] (6)

Derivation of the Activation Volume

In order to link the activation parameters to molecular aspects of the deformation process, we have determined the activation volume for very high strain rate [Mathematical Expression Omitted]. Figure 8 shows the dependence of the logarithm of [Mathematical Expression Omitted] on the ratio of the yield stress to absolute temperature. The slope of each line allow us to compute the activation volume at different temperatures. The values are gathered in Table 3.

Discussion

The measured activation volume, constant regardless of temperature (except at -30 [degrees] C), can rightly be identified to the true activation volume. It means that the yield behavior of the modified epoxy joint occurs for a same microstructural state in the corresponding temperature range. The term "microstructure" indicates here the characteristics of the arrangement of the structural units that compose the polymer after curing. The properties that define this arrangement are the activation enthalpy, the activation volume, and the entropy. Furthermore, it is worth noting that the mean value of the activation volume measured for the modified epoxy, referenced EP 1, is in a good agreement with the one determined by Escaig for dyciandamide-based DGEBA, but differs from the one determined for a dyciandamide-based DGEBU (Table 4). [V.sub.a] is then strongly sensitive to the local flexibility of the chains. These first results show clearly the power of the thermodynamic approach, related to the Somigliana dislocation model, to account for the behavior of polymers at high strain rates, in rapport with the chain flexibility.

[TABULAR DATA FOR TABLE 3 OMITTED]

On the other hand, at a low temperature, -30 [degrees] C, the deformation is confined in a reduced volume, the molecular motions are restricted and the entropy of the polymer decreases (structural modification of the polymer). So, the latter presents a stronger resistance to the deformation: [[Tau].sub.[Alpha]] increases (Table 5) and the high strain rates involved, promote a very local heating, which generates a nonlinear viscoelastic behavior (11, 12). Therefore, in accordance with the results obtained from the Bauwens approach, the fact that Ta increases notably and that the activation volume decreases, means a deformation process that is very heterogeneous at -30 [degrees] C. Both molecular models are then no longer valid.

Derivation of the Activation Energy

To obtain the true activation energy from Eq 6, we have fixed a strain rate sufficiently high [Mathematical Expression Omitted] and then determined graphically the slope of the line representing the variation of the yield stress versus the temperature [ILLUSTRATION FOR FIGURE 9 OMITTED]. The activation energy, directly proportional to the temperature, reveals a thermal activated deformation mechanism:

[Delta][H.sub.a](kJ/mote) = 12 RT (with [V.sub.a] = 165 [A.sup.3]) (7)

The dependence of the activation energy on temperature is shown in Fig. 10. We can note that, at 353 [degrees] K, the activation energy corresponds to that of the [Beta] transition. This result is in good agreement with the Eyring-Bauwens approach, which showed that, at 353 [degrees] K, the change in the deformation process of the polymer (from the [Alpha] mode to the [Alpha] + [Beta] model occurs effectively at a strain rate of [Mathematical Expression Omitted] (Table 1 and [ILLUSTRATION FOR FIGURE 4 OMITTED]).

Discussion

In his approach, Escaig defines the stress, [[Tau].sup.*] = [Tau] [[Tau].sub.i], as the effective stress to overcome the enthalpic barrier related to changes of the molecular conformations which control the deformation. In agreement with the formalism of Escaig, [[Tau].sub.[Alpha]], the stress required to stretch the mean chains, and so to reorganize the macromolecular network, can rightly be identified to [[Tau].sub.i]. Therefore, Eq 6 can be written as follows:

[Delta][H.sub.o] = [Delta][H.sub.a] + ([Tau] - [[Tau].sub.[Alpha]])[V.sub.a] (8)

Thus, as long as the temperature is below that corresponding to the presence of the [Beta] relaxation peak, the mobilization of the [Beta] molecular motions are preponderant and a significant mechanical stress is required to activate these local motions, [[Tau].sup.*] = [Tau] - [[Tau].sub.[Alpha]] = [[Tau].sub.[Beta]].

[TABULAR DATA FOR TABLE 5 OMITTED]

This stress decreases (whereas [Delta][H.sub.a] increases) as the temperature of the test is increased [ILLUSTRATION FOR FIGURE 10 OMITTED].

When the temperature reaches the critical temperature [T.sub.[Beta]], 353 [degrees] K for the given strain rate [Mathematical Expression Omitted], the thermal fluctuations are sufficient to activate the [Beta] motions without any mechanical energy contribution: [Tau] = [[Tau].sub.[Alpha]]. In this case, [Delta][H.sub.a] = [Delta][H.sub.o] and [Delta][H.sub.o] is nothing more than the [Beta] activation energy [Q.sub.[Beta]].

So, Eq 8 clearly shows the agreement between the thermodynamic model of Escaig and the Eyring-Bauwens approach, to account for the behavior of the modified epoxy joint at high strain rates in terms of molecular motions linked to the [Beta] relaxation. This agreement is well shown in Fig. 11, which shows the transition point in the yield of the modified epoxy joint, expressed in terms of a critical strain rate (Eyring-Bauwens approach) or in terms of a critical temperature (Escaig model).

In his approach (13), Perez managed to correlate the activation volume with the activation energy [Q.sub.[Beta]]:

[V.sub.ath] = 3[Q.sub.[Beta]] [(1 - [Tau]/[[Tau].sub.o]).sup.1/2]/2[Chi] [[Tau].sub.o] (9)

where [[Tau].sub.o] is the stress required to activate the molecular motions without any thermal energy contribution, and [Chi] is a parameter of correlation between 0 and 1.

CONCLUSION

The aim of this study was to show the excellent impact behavior of a bonded joint by taking into account the viscoelastic properties of the adhesive. The investigation of a modified epoxy joint properties, at high strain rates and over a wide range of temperatures, has highlighted the good correlation between high impact resistance and the presence of the [Beta] mechanical loss peak in the range of the strain rates explored.

Assuming that high polymers exhibit purely viscous yield, the Eyring-Bauwens approach and the Escaig model explain the strain rate sensitivity of the yield stress in terms of a difference in relaxation times at low strain rates (the [Alpha] relaxation) and high strain rates (the [Beta] relaxation).

Both models predict the presence of a transition in the yield of the adhesive joint, expressed in term of a critical strain rate [Mathematical Expression Omitted] or a critical temperature [T.sub.[Beta]] [Mathematical Expression Omitted].

The Escaig model has the advantage of containing two relevant parameters: the internal stress [[Tau].sub.i] (assimilate to [[Tau].sub.[Alpha]] for the concerned epoxy adhesive), which measures the microstructural changes of the polymer, and the activation volume [V.sub.a], which accounts for macromolecular chain flexibility.

But if the evolution of the yield stress is explainable in the range 24 [degrees] C to 80 [degrees] C, at low temperature (-30 [degrees] C), both models are no longer valid. The evolution of the micromolecular structure with the formation of local shear bands causes heterogeneous deformation, which cannot be simply interpreted from these models.

ACKNOWLEDGMENTS

The authors would like to thank SOLLAC, RENAULT, PEUGEOT S.A., CECA S.A., and the Ministry of Universities and Research for supporting this research program and the "Agence Nationale de la Recherche et de la Technologie" for the grant associated to the PhD thesis of F. Cayssials, and the Regional Council Aquitaine for its support in the experimental facilities.

REFERENCES

1. J. C. Bauwens, J. Mater. Sci., 7, 577 (1972).

2. T. Ree and H. Eyring, in Rheology, Vol. 2, Chap. 3, R. Eirich, ed., Academic Press, New York (1958).

3. J. M. Lefebvre and B. Escaig, Polymer, 34, 518 (1993).

4. D. R. Ireland, Instrumented Impact Testing, ASTM STP 563, pp. 3-29 (1974).

5. J. L. Lataillade, F. Cayssials, D. Crapotte, and C. Keisler, Recent Advances in Experimental Mechanics, Vol. 2, pp. 1295-1300, 10th International Conference on Experimental Mechanics, Lisbon (1994).

6. A. Molinari and Ch. G'Sell, in Introduction & la Mecanique des Polymeres, pp. 321-44, Ch. G'Sell and J. M. Haudin (1995).

7. A. M. North, Int. Rev. Sci., 8, 1 (1975).

8. S. Cukierman, J. L. Halary, and L. Monnerie, Polym. Eng. Sci., 31, 1476 (1991).

9. F. Cayssials, thesis, ENSAM Bordeaux, France (1995).

10. J. F. Gerard, S. J. Andrews, and C. W. Macosko. Polym. Compos., 11, 90 (1990).

11. J. Perez and J. M. Lefebvre, in Introduction a la Mecanique des Polymeres, pp. 289-318, Ch. G'Sell and J. M. Haudin, eds. (1995).

12. L. Ladouce, thesis, INSA Lyon, France (1995).

13. J. Perez, in Physique et mecanique des polymeres amorphes, Lavoisier, Paris (1992).

The impact behavior of new, lightweight bonded structures requires careful examination. Adhesives, being polymeric materials, have properties that are usually rate dependent, because of their viscoelasticity. It is by no means certain that their performance under static conditions will be the same as under dynamic ones. There is, therefore, the need to give designers some confidence that under impact conditions, bonded joints will maintain their structural integrity. Unfortunately, few studies have been carried out to investigate properties of bonded joints at medium and high strain rates. For this purpose, a new experimental setup based on the Charpy technique (the inertia wheel) was used to characterize the impact resistance of structural adhesive joints ([Mathematical Expression Omitted] to [10.sup.4] [s.sup.-1]) under shear loading. The study was completed by means of static tests and temperature tests (-30, 40, 60, 80 [degrees] C). We found there was a critical strain rate [Mathematical Expression Omitted] (T) (which is a function of temperature), above which the joint becomes more resistant.

This change of behavior, which affects not only the yield stress but also the failure energy, was successfully explained using the Bauwens approach (1) which consists of a modification of the Ree-Eyring (2) theory.

In this paper, we show that the Escaig model (3) confirms the role played by the [Beta] mechanical loss peak on the strain rate sensitivity of the yield stress. It was indeed possible to derive the activation energy associated with the [Beta] transition of our modified epoxy resin from both models, which is found to be in good agreement with the values given in the literature. The Escaig model has the advantage of containing two relevant parameters: the volume activation [V.sub.a], which reflects the local flexibility of the macromolecular chains, and a nonlinear parameter [[Tau].sub.i], which determines the effect of changes in the polymer structure. Furthermore, the evolution of both parameters allows us to explain accurately the limitation of these models at low temperatures (-30 [degrees] C).

EXPERIMENTAL PROCEDURES

Materials

The two structural adhesives investigated were industrial one-part adhesives produced by CECA Adhesifs: An unmodified epoxy resin and a modified one. The two components of the unmodified resin are i) the epoxy resin (a blend of different DGEBA by-products) and ii) the curing agent (Dicyandiamine). The modified epoxy resin was obtained by incorporating in the resin a block copolymer modifier (commercially secret) and fillers (aluminium, amorphous silica, and calcium carbonate). The substrate was a galvanized steel sheet (thickness: 1.5 mm). Before bonding, steel sheets were simply cleaned in an ethyl acetate bath.

Sample

The specimens are made of two galvanized steel sheets bonded by epoxy resin [ILLUSTRATION FOR FIGURE 1 OMITTED]. The two steel sheets are clamped together with PTFE spacers at each side of the substrate to establish its nominal thickness (0.5 mm).

The adhesive was cured by heating in an oven, 30 min at 180 [degrees] C. To carry out loading using the impact test system, the sample was fixed onto rigid supports. Fixation was obtained by bonding with a cyanacrylate adhesive applied to the whole surface of the steel area, which is sufficient to strongly attach it to the support [ILLUSTRATION FOR FIGURE 2 OMITTED].

High Strain Rate Tests

The high strain rate tests were carried out on a specific experimental setup, the so-called "Inertia Wheel" [ILLUSTRATION FOR FIGURE 3 OMITTED]. This new device based on the inertia technique is very suitable; the tangential velocity of the wheel can be adjusted very accurately (from 1 m/s to 20 m/s). When the wheel reaches the nominal tangential velocity, the specimen rotates just before the hammer fixed on the wheel by means of a pendulum mechanism activated by a pneumatic jack [ILLUSTRATION FOR FIGURE 4 OMITTED].

A charge amplifier converts the electric charge supplied by the piezoelectric transducer into a voltage signal which is proportional to the applied force. The signal is stored in a numerical oscilloscope. A Zimmer OHG extensometer (bandwidth: 200 kHz) is used to follow the relative displacement of the two substrates by means of two marks stuck on each support [ILLUSTRATION FOR FIGURE 5 OMITTED]. The high energy stored by the wheel compared to the energy required to break a joint allows us to obtain a strain rate that is nearly constant throughout the test [ILLUSTRATION FOR FIGURE 6 OMITTED]. Since the frequency response of the quartz washer is 40 kHz and the smallest time to failure in any test is [approximately]30 [[micro]seconds], this system is obviously adequate (4, 5).

Since the displacements due to the rigid supports are not negligible, they must be considered. For this reason, further tests with the supports alone were carried out and then, the apparatus compliance is taken into account [ILLUSTRATION FOR FIGURE 6 OMITTED].

Figure 6 shows that the polymer does not exhibit a purely linear behavior. The maximum point on the curve is sufficiently remarkable to consider it the yield point to account for the pre-plastic deformation of the epoxy, related to the multiplication of shears bands, which self-accelerate (6). Thus, the yield stress and the critical energy can readily be obtained from the following equations:

[[Tau].sub.y] = [F.sub.max]/[A.sub.joint] (1)

[E.sub.c] = [V.sub.ham]/[A.sub.joint] [[integral of] [F.sub.1](t)dt between limits [t.sub.f] and 0 - [integral of] [F.sub.2](t)dt between limits [t.sub.s] and 0] (2)

Where:

[V.sub.ham]: velocity of the hammer

[V.sub.ham] [multiplied by] [t.sub.f]: displacement of the adhesive joint and the supports.

[V.sub.ham] [multiplied by] [t.sub.s]: displacement due to the only supports.

[A.sub.joint]: area of the joint

Tests at Lower Strain Rates

The shearing tests at lower speeds were carried out using an Instron test machine ([10.sup.-2] to [10.sup.1] [s.sup.-1]) and a hydraulic tensile machine ([10.sup.2] to [10.sup.3] [s.sup.-1]).

RESULTS AND DISCUSSION

We successfully applied the Bauwens approach to describe the behavior of the modified epoxy joint (0.5 mm thick joint) over a wide range of strain rates and temperatures. The important results concerning this approach are summarized in a first step. Then, to confirm the important role played by the viscoelastic properties of the adhesive on the strain rate sensitivity of the yield stress, we extended the study by applying the Escaig model, which has the advantage of taking account of the structural modifications of the polymer. That will be the subject of the last part of this contribution.

The Eyring-Bauwens Model

If the polymer is capable of exhibiting a secondary transition [Beta], (associated to the local relaxation modes of the macromolecular structure or the relaxation of the side groups), Bauwens showed that at high strain rates, the variation of the yield stress with the strain rate can be described by the generalized theory proposed by Ree-Eyring (2):

[Mathematical Expression Omitted] (3)

where T is the absolute temperature and R is the universal gas constant. The parameters [A.sub.[Alpha]], [C.sub.[Alpha]], [A.sub.[Beta]], [C.sub.[Beta]] are characteristics of the material and [Q.sub.[Alpha]], [Q.sub.[Beta]] are the energy barriers of respectively the [Alpha] relaxation and the [Beta] relaxation. These barrier energies have to be overcome so that a segment macromolecular can jump from one equilibrium position to another. This model considers that below a critical strain rate [Mathematical Expression Omitted], only the activation of the main relaxation ([Alpha] relaxation), associated to the relaxation of the crosslinked polymer network, requires a mechanical energy. On the other hand, above [Mathematical Expression Omitted], both [Alpha] and [Beta] need to be surmounted. In epoxy systems, this [Beta] peak is related to motions of the ether linkage segments in the epoxy component of the resin (7):

[Mathematical Expression Omitted]

The secondary transition ([Beta]) of the modified epoxy has been located at -60 [degrees] C, from dynamic mechanical spectroscopy measurements at 1 Hz.

Comparison Between Bauwens Model and Experimental Results

Figure 7 shows the ratio of the yield stress to absolute temperature versus the logarithm of [Mathematical Expression Omitted] at different temperatures, for a 0.5-mm-thick joint. The full lines plotted in Fig. 7 correspond to the theorical variation using the following Eyring equation:

[Mathematical Expression Omitted] (4)

with [X.sub.[Alpha]] = ln(2[C.sub.[Alpha]]) + [Q.sub.[Alpha]]/RT and [Mathematical Expression Omitted]

For each temperature, the parameters of Eq 4 are fitted to the experimental results, allowing us to determine [Q.sub.[Alpha]], [Q.sub.[Beta]], [C.sub.[Alpha]] and [C.sub.[Beta]] (Table 1).

Discussion

It is worth noting that the parameters are constant whatever the temperature (except for -30 [degrees] C). Furthermore, the [Beta] activation energy agrees with that given in the literature for amine-based DGEBA (8) (Table 2). To check the accuracy of this interpretation, a Fourier analysis (9) was also performed to obtain the complete frequency spectrum of the impact pulse corresponding to [Mathematical Expression Omitted]. Every [Mathematical Expression Omitted] is linked to a limit frequency [f.sub.c](T). These values were used in the Arrhenius time-temperature conversion, allowing us to calculate an activation energy [Q.sub.[Beta]] close to those found by other authors or calculated from the Eyring equation, reinforcing the Bauwens approach (Table 2). Therefore, in the temperature range from 24 to 80 [degrees] C, the yield behavior of the modified epoxy joint, above [Mathematical Expression Omitted], may be described by the sum of the partial stresses [[Tau].sub.[Alpha]] and [[Tau].sub.[Beta]] required to free the different kinds of molecular motions implied in the deformation process.

Dynamic mechanical analysis, performed at 1 Hz, revealed the existence of the [Beta] transition for the modified epoxy adhesive at -60 [degrees] C. But let us note that even though the secondary transition is located at a low temperature does not mean that, at room temperature, the local molecular motions are free to move. For instance, an impact stress pulse corresponds to high-frequency measurement, which, by the time-temperature equivalence principle, has the effect of shifting the secondary transition to higher temperatures.

In that way, depending on the temperature and the strain rate, the molecular motions of the [Beta] relaxation [TABULAR DATA FOR TABLE 1 OMITTED] may be frozen or not. The point transition is given by [Mathematical Expression Omitted].

Table 2. Computed Activation Energies Compared With Those Given in Literature. Values Value From The Activation Values in From This Arrhenius Energies Literature (10) Work Relation [Q.sub.[Alpha]] 400-800 kJ/mole 470 kJ/mole - [Q.sub.[Beta]] 42-60 kJ/mole 38 kJ/mole 37.5 kJ/mole

On the other hand, the significant diminution of An and [C.sub.[Beta]] at -30 [degrees] C is likely to arise as a result of microstructural changes in the polymer: at high strain rates and low temperatures, the deformation is confined to a reduced volume, which promotes a concentration of adiabatic shear bands and a decrease of the yield stress (characterized by a decrease of [A.sub.[Beta]]). The deformation is no longer homogeneous (nonlinear behavior), and the parameter [C.sub.[Beta]] (which contains a frequency factor), related to the relaxation time [[Tau].sub.[Beta]], becomes sensitive to temperature (2, 6).

Unfortunately, it is difficult to explain this phenomenon from the Eyring-Bauwens model since it does not account for the structural modifications of the polymer, and heterogeneous fields of deformation.

The Escaig model (3), developed in the eighties, uses a complex formalism, but has the advantage of being based on a nonlinear approach. It is thus much more appropriate for describing the inelastic deformation of polymers. The recent results obtained from this model are reported below.

The Escaig Model

The yielding behavior of thermosets is reviewed in terms of a thermal activated dislocation propagation mechanism. Entanglements are viewed as a local barriers [Delta][H.sub.o] to be overcome with the help of thermal fluctuations and mechanical work.

The jumping of the local barrier, required for the propagation of a dislocation, may be written as follows:

[Delta][H.sub.o] = [Delta][H.sub.a] + ([Tau] - [[Tau].sub.t])[V.sub.a] (5)

Where:

* [V.sub.a] is associated to the spatial extension of the shear bands located where the molecular motions are very important.

* [[Tau].sub.i] is an internal stress that accounts for the entropic effects due to the chain elongation. It is thus directly proportional to [[Tau].sub.[Alpha]].

* [Delta][H.sub.a] is the activation energy, which accounts for the thermal fluctuations.

[V.sub.a] and [Delta][H.sub.a] characterize, respectively, the stress and the temperature sensitivity of the strain rate. These important activation parameters make it possible to obtain useful information about the polymer deformation mechanisms. Indeed, if [V.sub.a] is independent of the stress and the temperature, it is possible to relate the activation volume and the activation enthalpy to parameters measured by experiment (10):

[Mathematical Expression Omitted] (6)

Derivation of the Activation Volume

In order to link the activation parameters to molecular aspects of the deformation process, we have determined the activation volume for very high strain rate [Mathematical Expression Omitted]. Figure 8 shows the dependence of the logarithm of [Mathematical Expression Omitted] on the ratio of the yield stress to absolute temperature. The slope of each line allow us to compute the activation volume at different temperatures. The values are gathered in Table 3.

Discussion

The measured activation volume, constant regardless of temperature (except at -30 [degrees] C), can rightly be identified to the true activation volume. It means that the yield behavior of the modified epoxy joint occurs for a same microstructural state in the corresponding temperature range. The term "microstructure" indicates here the characteristics of the arrangement of the structural units that compose the polymer after curing. The properties that define this arrangement are the activation enthalpy, the activation volume, and the entropy. Furthermore, it is worth noting that the mean value of the activation volume measured for the modified epoxy, referenced EP 1, is in a good agreement with the one determined by Escaig for dyciandamide-based DGEBA, but differs from the one determined for a dyciandamide-based DGEBU (Table 4). [V.sub.a] is then strongly sensitive to the local flexibility of the chains. These first results show clearly the power of the thermodynamic approach, related to the Somigliana dislocation model, to account for the behavior of polymers at high strain rates, in rapport with the chain flexibility.

[TABULAR DATA FOR TABLE 3 OMITTED]

Table 4. Activation Volume Values Measured From our Experience and From Escaig's Experience for Dyciandamide-Based DGEBA. EP 1 DGEBA - DDM DGEBU - DDM [V.sub.a] ([A.sub.3]) 165 [+ or -] 4 170 70

On the other hand, at a low temperature, -30 [degrees] C, the deformation is confined in a reduced volume, the molecular motions are restricted and the entropy of the polymer decreases (structural modification of the polymer). So, the latter presents a stronger resistance to the deformation: [[Tau].sub.[Alpha]] increases (Table 5) and the high strain rates involved, promote a very local heating, which generates a nonlinear viscoelastic behavior (11, 12). Therefore, in accordance with the results obtained from the Bauwens approach, the fact that Ta increases notably and that the activation volume decreases, means a deformation process that is very heterogeneous at -30 [degrees] C. Both molecular models are then no longer valid.

Derivation of the Activation Energy

To obtain the true activation energy from Eq 6, we have fixed a strain rate sufficiently high [Mathematical Expression Omitted] and then determined graphically the slope of the line representing the variation of the yield stress versus the temperature [ILLUSTRATION FOR FIGURE 9 OMITTED]. The activation energy, directly proportional to the temperature, reveals a thermal activated deformation mechanism:

[Delta][H.sub.a](kJ/mote) = 12 RT (with [V.sub.a] = 165 [A.sup.3]) (7)

The dependence of the activation energy on temperature is shown in Fig. 10. We can note that, at 353 [degrees] K, the activation energy corresponds to that of the [Beta] transition. This result is in good agreement with the Eyring-Bauwens approach, which showed that, at 353 [degrees] K, the change in the deformation process of the polymer (from the [Alpha] mode to the [Alpha] + [Beta] model occurs effectively at a strain rate of [Mathematical Expression Omitted] (Table 1 and [ILLUSTRATION FOR FIGURE 4 OMITTED]).

Discussion

In his approach, Escaig defines the stress, [[Tau].sup.*] = [Tau] [[Tau].sub.i], as the effective stress to overcome the enthalpic barrier related to changes of the molecular conformations which control the deformation. In agreement with the formalism of Escaig, [[Tau].sub.[Alpha]], the stress required to stretch the mean chains, and so to reorganize the macromolecular network, can rightly be identified to [[Tau].sub.i]. Therefore, Eq 6 can be written as follows:

[Delta][H.sub.o] = [Delta][H.sub.a] + ([Tau] - [[Tau].sub.[Alpha]])[V.sub.a] (8)

Thus, as long as the temperature is below that corresponding to the presence of the [Beta] relaxation peak, the mobilization of the [Beta] molecular motions are preponderant and a significant mechanical stress is required to activate these local motions, [[Tau].sup.*] = [Tau] - [[Tau].sub.[Alpha]] = [[Tau].sub.[Beta]].

[TABULAR DATA FOR TABLE 5 OMITTED]

This stress decreases (whereas [Delta][H.sub.a] increases) as the temperature of the test is increased [ILLUSTRATION FOR FIGURE 10 OMITTED].

When the temperature reaches the critical temperature [T.sub.[Beta]], 353 [degrees] K for the given strain rate [Mathematical Expression Omitted], the thermal fluctuations are sufficient to activate the [Beta] motions without any mechanical energy contribution: [Tau] = [[Tau].sub.[Alpha]]. In this case, [Delta][H.sub.a] = [Delta][H.sub.o] and [Delta][H.sub.o] is nothing more than the [Beta] activation energy [Q.sub.[Beta]].

So, Eq 8 clearly shows the agreement between the thermodynamic model of Escaig and the Eyring-Bauwens approach, to account for the behavior of the modified epoxy joint at high strain rates in terms of molecular motions linked to the [Beta] relaxation. This agreement is well shown in Fig. 11, which shows the transition point in the yield of the modified epoxy joint, expressed in terms of a critical strain rate (Eyring-Bauwens approach) or in terms of a critical temperature (Escaig model).

In his approach (13), Perez managed to correlate the activation volume with the activation energy [Q.sub.[Beta]]:

[V.sub.ath] = 3[Q.sub.[Beta]] [(1 - [Tau]/[[Tau].sub.o]).sup.1/2]/2[Chi] [[Tau].sub.o] (9)

where [[Tau].sub.o] is the stress required to activate the molecular motions without any thermal energy contribution, and [Chi] is a parameter of correlation between 0 and 1.

CONCLUSION

The aim of this study was to show the excellent impact behavior of a bonded joint by taking into account the viscoelastic properties of the adhesive. The investigation of a modified epoxy joint properties, at high strain rates and over a wide range of temperatures, has highlighted the good correlation between high impact resistance and the presence of the [Beta] mechanical loss peak in the range of the strain rates explored.

Assuming that high polymers exhibit purely viscous yield, the Eyring-Bauwens approach and the Escaig model explain the strain rate sensitivity of the yield stress in terms of a difference in relaxation times at low strain rates (the [Alpha] relaxation) and high strain rates (the [Beta] relaxation).

Both models predict the presence of a transition in the yield of the adhesive joint, expressed in term of a critical strain rate [Mathematical Expression Omitted] or a critical temperature [T.sub.[Beta]] [Mathematical Expression Omitted].

The Escaig model has the advantage of containing two relevant parameters: the internal stress [[Tau].sub.i] (assimilate to [[Tau].sub.[Alpha]] for the concerned epoxy adhesive), which measures the microstructural changes of the polymer, and the activation volume [V.sub.a], which accounts for macromolecular chain flexibility.

But if the evolution of the yield stress is explainable in the range 24 [degrees] C to 80 [degrees] C, at low temperature (-30 [degrees] C), both models are no longer valid. The evolution of the micromolecular structure with the formation of local shear bands causes heterogeneous deformation, which cannot be simply interpreted from these models.

ACKNOWLEDGMENTS

The authors would like to thank SOLLAC, RENAULT, PEUGEOT S.A., CECA S.A., and the Ministry of Universities and Research for supporting this research program and the "Agence Nationale de la Recherche et de la Technologie" for the grant associated to the PhD thesis of F. Cayssials, and the Regional Council Aquitaine for its support in the experimental facilities.

REFERENCES

1. J. C. Bauwens, J. Mater. Sci., 7, 577 (1972).

2. T. Ree and H. Eyring, in Rheology, Vol. 2, Chap. 3, R. Eirich, ed., Academic Press, New York (1958).

3. J. M. Lefebvre and B. Escaig, Polymer, 34, 518 (1993).

4. D. R. Ireland, Instrumented Impact Testing, ASTM STP 563, pp. 3-29 (1974).

5. J. L. Lataillade, F. Cayssials, D. Crapotte, and C. Keisler, Recent Advances in Experimental Mechanics, Vol. 2, pp. 1295-1300, 10th International Conference on Experimental Mechanics, Lisbon (1994).

6. A. Molinari and Ch. G'Sell, in Introduction & la Mecanique des Polymeres, pp. 321-44, Ch. G'Sell and J. M. Haudin (1995).

7. A. M. North, Int. Rev. Sci., 8, 1 (1975).

8. S. Cukierman, J. L. Halary, and L. Monnerie, Polym. Eng. Sci., 31, 1476 (1991).

9. F. Cayssials, thesis, ENSAM Bordeaux, France (1995).

10. J. F. Gerard, S. J. Andrews, and C. W. Macosko. Polym. Compos., 11, 90 (1990).

11. J. Perez and J. M. Lefebvre, in Introduction a la Mecanique des Polymeres, pp. 289-318, Ch. G'Sell and J. M. Haudin, eds. (1995).

12. L. Ladouce, thesis, INSA Lyon, France (1995).

13. J. Perez, in Physique et mecanique des polymeres amorphes, Lavoisier, Paris (1992).

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Title Annotation: | French Research on Structural Properties of Polymers |
---|---|

Author: | Lataillade, J.L.; Cayssials, F. |

Publication: | Polymer Engineering and Science |

Date: | Oct 1, 1997 |

Words: | 3634 |

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