Observation of Yield in Triaxial Deformation of Glassy Polymers

Jae Woo Kim, Grigori A. Medvedev and James M. Caruthers1

School of Chemical EngineeringPurdue University

Abstract

Yield in amorphous glassy polymers has been studied in shear and uniaxial tension/ compression,

where the volume change is relatively small and the deviatoric components of the strain/stress tensor are

dominate. In order to study volumetrically driven yield in a different geometry, a specially designed

experimental set-up was developed to create a longitudinal deformation that is dilatationally dominated.

The longitudinal stress response of an epoxy resin was measured at temperatures just below Tg, where for

the first time yield of a glassy polymer is observed in a nearly triaxial extension deformation. The yield

behavior in uniaxial tension and compression was also determined, where all the yield data can be

described with pressure-modified von Mises yield criterion but now for much wider pressure range than

has been previously accessible.

Keywords

yield; triaxial deformation; epoxy resin

1 Corresponding Author: 480 Stadium Mall Drive, West Lafayette, IN 47907 caruther@ecn.purdue.edu

1

1. Introduction

An understanding of the response of glassy polymers to various types of deformations is critical for

both fundamental understanding of the nature of the glassy state and wide-spread use of these materials in

important engineering applications. The large deformation behavior of glassy polymers is typically

measured in uniaxial tension and compression, where both brittle and ductile behavior has been observed.

[1-3] The ductile response is characterized by the significant drop in the tangent modulus and sometimes

is followed by post-yield softening with eventual hardening, where the features of the stress-strain curve

depend on the thermal and mechanical history, i.e. temperature, aging time, strain rate, etc.[1, 4-6] Yield

behavior of glassy polymers has also been observed in shear.[7, 8] In both uniaxial and shear

deformations, the associated volume change is small, reaching a maximum value of approximately 0.5%

at the yield point[9]; thus, uniaxial and shear experiments primarily probe the effect of the deviatoric

contribution of the strain/stress tensor.[10] However, in many applications polymers experience multi-

axial deformation states with a significant dilatational component, e.g. in continuous fiber composites the

polymer matrix deformations can range from predominantly deviatoric to predominantly dilatational

depending on location in the composite.[11] In order to de-convolute the deviatoric and dilatational

contributions in a multi-axial deformation a variety of deformation geometries have been investigated

including uniaxial extension, uniaxial compression, simple shear, plane strain compression, biaxial

extension and thin-walled hollow cylinder torsion tests[6-8, 12-16]; however, the deviatoric contribution

dominates in all of these deformations. More recently, a triaxial strain/stress state with significant

dilatation was investigated by Berglund, et al. for an epoxy polymer using the poker chip method, where

brittle failure was observed.[17, 18] To the best of our knowledge, ductile failure has not been reported

for a glassy polymer in a dilatationally dominated deformation.

For metals yield is defined as the stress at which a material begins to deform plastically and is well

explained by the irreversible slippage of dislocations along neighboring crystalline planes.[4, 6] In the

2

three-dimensional principal stress space, the locus of yield points forms a yield surface, which is used in

engineering design as an upper limit of the load that can be applied in a complex deformation field. In

contrast, yield of glassy polymers is a nonlinear time-dependent phenomenon that is fully reversible for

thermosetting systems by heating above the material's glass transition temperature.[4, 19] The underlying

physical mechanism for yield in glassy polymers is currently a matter of significant debate. Both

nonlinear viscoelastic[10, 20-22] and nonlinear viscoplastic[23-26] constitutive models have been

proposed to describe the experimentally observed yield behavior, where the rate of viscoelastic relaxation

(or equivalently the viscosity in a viscoplastic model) is deformation dependent. Constitutive models for

describing nonlinear relaxation in the glassy state, including yield, have been developed, where the

deformation dependent rate of relaxation has been postulated to depend upon free volume, stress, strain,

configurational entropy and configurational energy.[10, 20-22, 27, 28] Alternatively, plasticity based

constitutive models have been developed to describe the temperature and strain rate dependence of the

yield stress using a modified Eyring model,[24] where the plastic viscosity depends upon the deviatoric

component of the stress.[23, 24] In order to evaluate these various constitutive hypotheses concerning the

physical origin of yield, nonlinear deformation experiments have been performed at various strain rates,

temperatures, and thermal histories.[1, 5, 30] However, these studies have all been deviatorically

dominated deformations, where a more discriminating probe of the various postulates is how the

nonlinear time-dependent response, including yield, depends upon a dilatational vs. deviatoric

deformation, e.g. for a free volume model there would be minimal deformation induced nonlinearity in

shear vs. considerable nonlinearity in a uniaxial deformation. Thus, the use of dilatationally dominated

deformations can provide valuable information concerning the underlying physical processes that control

nonlinear viscoelastic relaxation and yield.

From a purely phenomenological perspective that is independent of the underlying physics, various

continuum postulates have been proposed to describe a time independent yield surface.[31, 32] For

example, the von Mises yield criterion assumes that yield occurs when the second deviatoric stress

3

invariant reaches a critical value. The von Mises yield criterion was originally used for ductile crystalline

materials such as metals, where yielding is essentially independent of the hydrostatic component of stress.

In the space of the principal components of the stress tensor, the von Mises yield criterion results in a

cylindrical-shaped yield surface aligned along the 111 direction. Since it has only one unknown

parameter, the von Mises yield surface for a given material can be fully characterized with just a single

experiment, for instance a uniaxial extension test. In contrast, yielding in glassy polymers is sensitive to

hydrostatic pressure.[7, 15, 29, 33-36] Bauwens states that the ratio of uniaxial compressive to tensile

yield (at the same temperature and strain rate) is 1.3,[7] which clearly shows the influence of the

hydrostatic pressure on the yield surface. The linear dependence of the uniaxial yield stress with

hydrostatic pressure has been observed in both semi-crystalline polymers (i.e. polyethylene[8],

polyoxymethylene,[37] polytetrafluoroethylene[35], poly(ethylene teraphthalate[8]) and amorphous

glassy polymers (i.e. polycarbonate[35], poly(ethylene teraphthalate)[15], poly(methyl methacrylate)[14],

polyvinyl chloride[7] and polystyrene[15]). In order to incorporate the effect of pressure on the yield

surface, a modified von Mises yield criterion is employed, where the mean octahedral shear stress, τoct, at

yield is assumed to be a linear function of pressure.[6, 15] Specifically,

(1)

where the empirical parameter μ is the so called coefficient-of-friction. In the principle stress space the

pressure modified von-Mises criterion results in a cone with an axis along the line. For

metals the physics behind a von Mises yield criterion is fairly well understood;[31, 32] however, for

amorphous glassy polymers the pressure-modified von Mises yield criterion is a phenomenological

assumption. In order to validate the empirical yield criterion given in Eqn. (1) three independent yield

experiments are required – two experiments to obtain the values of the parameters and μ, and a third to

evaluate the prediction of Eqn. (1) with an independently measured value of yield stress. To date, the

pressure modified von Mises yield criterion has been tested in various sets of experiments that employ a

4

combination of uniaxial tension, uniaxial compression, and either shear, plane strain compression with or

without applied tension, thin-walled hollow cylinder tension with internal pressure, biaxial tension and/or

combined tension-torsion.[7, 14, 15, 36, 38] In all cases the modified von Mises yield criterion was

validated; however, all the deformation conditions used in all these studies are deviatorically dominated,

where the hydrostatic stress remains within a narrow range near ambient pressure. The yield behavior

under the dilatation-dominant stress conditions such as occurs in the triaxial deformation is still in

question, since it has not been studied in well-defined laboratory experiments. In summary, using the

pressure modified von Mises criterion to predict the yield stress in a dilatationally dominated deformation

like tri-axial extension requires a large extrapolation to a region of stress space, where the validity of the

criterion itself has not been established.

In this communication we will report the observation of yield in a longitudinal stress state, where the

dilatational contribution is dominant. The longitudinal yield data in conjunction with yield data in

uniaxial tension and compression for the same material under the same thermal conditions/histories will

enable a critical examination of the pressure modified von Mises yield criterion. The rest of this paper is

organized as follows: in the Material Synthesis and Characterization section we will briefly describe (i)

the synthesis of the glassy epoxy resin system and (ii) various experiments used to characterize the

thermo-mechanical properties of the resin. In the next section of the paper, we will describe a novel

longitudinal deformation testing rig and provide data that confirm the accuracy of this specific

longitudinal deformation experiment. In the Results section (i) the mechanical response of the epoxy resin

will be reported, including the observation of longitudinal yield when the material is in the glassy state

just below the glass transition temperature, Tg, and (ii) the pressure modified von Mises criterion will be

critically evaluated using the yield stress measured in uniaxial extension, uniaxial compression and

longitudinal extension. In the Discussion Section we will consider the implications of the experimentally

observed longitudinal yield with respect to understanding and modeling of the yield phenomenon in

glassy polymers.

5

2. Material Synthesis and Characterization

The materials used in this study are a series of epoxy systems, where the most extensive data set was

obtained for neopentyl glycol diglycidyl ether (DGENG) cured with 4,4’-methylenedianiline (MDA)

system, where chemical structures of the species are shown in Figure 1. All the chemicals were used as

received without any further purification. Systems studied include DGENG (Miller-Stephenson Chemical

Co.) cured with 4,4’MDA (Sigma Aldrich); EPON825 (Miller-Stephenson Chemical Co.) cured with

3,3’-diaminophenyl sulphone, i.e. DDS; EPON825 (Miller-Stephenson Chemical Co.) cured with

aminoethyl piperazine, AEP, (Sigma Aldrich); and, DER736 (Dow Chemical Co.) cured with 3,3’-

diaminophenyl sulphone, DDS, (Sigma Aldrich). For all systems a stoichiometric ratio of epoxy to curing

agent was used except for the EPON825-AEP system, where a range of off-stoichiometric mixtures were

prepared. The synthetic route for the DGENG-4,4’MDA epoxy is as follows: DGENG was first heated to

60°C in order to lower its viscosity for easier processing and then placed for 1hr in a vacuum oven to

remove any residual volatiles. The 4,4’MDA curing agent was added to the DGENG liquid resin, where

the reaction mixture was maintained at 100°C using a silicone oil bath. The mixture was rigorously stirred

for several minutes and placed in the vacuum oven at 60°C for 30 minutes to remove air entrapped during

mixing. The polymerization reaction was then carried out between a pair of glass plates pretreated with

Multi-shieldTM mold release (ZYVAX), which were separated by a Teflon-coated O-ring. The temperature

profile for curing was 80°C for 4hrs followed by post-cure at 130°C for 3hrs. The use of a stoichiometric

ratio of epoxy to amine and the specific two-step thermal history allow the epoxy-amine addition

reactions to go to completion while suppressing the etherification side reaction.[2, 39] The resultant

1.5mm thick sheet of fully cross-linked epoxy was cut into desired dimensions using a surface grinder

with a diamond blade and then kept inside a desiccator before thermal or mechanical testing. Specimens

for the other epoxy systems were prepared in a similar manner, where the thermal histories are given in

6

Table 1. For PVT measurements a cylindrical plug was produced using a similar procedure, but where

the curing reaction was in a test tube.

Figure 1: Chemical structures of DGENG and 44’MDA

The glass transition temperature was measured using a Q2000 (TA Instruments) differential scanning

calorimeter, which was calibrated using nickel rod and indium standards. A small amount of epoxy (5 to

10mg) was loaded inside DSC pan and an inert atmosphere was maintained by flowing nitrogen. Before

each run the sample was heated to 160°C, which was well above the glass transition temperature, for 30

minutes to erase any previous thermal history. The sample was then cooled at 30°C/min to 30°C,

thermally equilibrated for 5 minutes, and then heated at 10°C/min to 160°C. The glass transition

temperature was determined by analyzing the heat flow during the heating ramp. Neither decomposition

nor additional curing was observed during the temperature scan.

The pressure-volume-temperature (PVT) behavior was measured using a Gnomix pressure

dilatometer. The specimen was machined into cylindrical shape that was 8mm in diameter and 10mm in

height. The specific volume of the sample at room temperature (25°C) was measured by Archimedes’

7

Table 1: Cure history for various epoxy resins

system mixing temperature and time

curing procedure1st step 2nd step

DGENG-44’MDA 100°C, 10 mins 80°C (4hrs) 130°C (3hrs)EPON825-33’DDS 185°C, 10 mins 120°C (4hrs) 200°C (3hrs)

EPON825-AEP RT, 10 mins 100°C (4hrs) 160°C (3hrs)DER736-33’DDS 185°C, 10 mins 120°C (4hrs) 180°C (3hrs)

principle using a top loading balance. After loading into the Gnomix dilatometer the density change was

measured using the following temperature-pressure history:

1. The specimen was equilibrated for 30 minutes at 160°C at 10MPa to erase the effects of the previous thermal history.

2. The specimen was cooled to 25°C at the rate of 1°C/min with the pressure maintained at 10MPa. 3. At 25°C the specimen was pressurized in the increment of 25MPa from 10MPa to 200MPa and

the specific volume was recorded at each pressure after 5 minutes for thermal equilibration.4. The pressure was then decreased back to 10MPa. 5. At 10MPa the temperature was increased by 15°C, the specimen was allowed to equilibrate at this

new temperature for 30 mins and then the pressure history described in 3 and 4 was repeated. 6. The temperature-pressure protocol described in 5 was repeated up through 160°C.

The empirical Tait equation-of-state was used to fit the data.[40, 41] The PVT response at atmospheric

pressure, i.e. 0.1MPa, was determined via extrapolation of the Tait equation.

The dynamic shear moduli were determined in torsion using ARES LS2 rheometer (TA Instruments)

with a rectangular specimen with dimensions of 1.5mm × 7.5mm × 30mm. The strain magnitude of

0.005 was used. The temperature dependence of the linear viscoelastic storage and loss moduli in shear

were determined at a frequency of 10 rad/s for temperatures between 30°C to 160°C, where after each

10oC temperature increase the specimen was allowed to thermally equilibrate for 10 minutes prior to the

measurement of the dynamic moduli. During the dynamic torsional test, the environmental chamber was

purged with nitrogen to prevent thermal degradation.

The uniaxial tensile and compressive stress-strain measurements were performed using an Instron

5567 testing machine with a 3119 thermal chamber. The uniaxial extension specimens were rectangular

(1.5mm × 7.5mm × 40mm) and were attached with pressure-controlled grips. The thermal chamber was

pre-heated to the testing temperature. The testing temperatures reported in this communication are the

actual temperatures near the specimen as measured by thermocouples, not the nominal temperatures set

by the instrument. The temperature history for the tensile specimen was: hold at T g+30°C for 30 mins in

an oven, quench to room temperature by taking it out of the oven, and finally thermally equilibrate the

test specimen in the Instron thermal chamber at the testing temperature for 30 minutes prior to

8

deformation. For the compression experiments cylindrical specimens with a diameter of approximately

9.8mm and a diameter-to-length ratio of 1:1.5 were used. The thermal history used for the compression

specimens was the same as that used for the tensile specimens. An Instron 2663-821 video extensometer

was used to measure the displacement in the uniaxial extension experiments, while in compression the

strain was determined from the crosshead displacement. In the compression experiments the machine

compliance correction method of Kalidindi et al. [42] was used.

3. Experimental Apparatus for Longitudinal Deformation

A novel experimental set-up has been developed for longitudinal deformation where dilatational

contribution is dominant. The specimen is a thin epoxy layer cured between the flat end faces of a pair of

2.54cm diameter 309 circular stainless steel rods separated with a 250m teflon spacer as shown

schematically in Fig. 2, where a silane coupling agent was applied to the steel surface to improve

interfacial adhesion with epoxy system. The detailed procedure for manufacturing the longitudinal test

specimen is:

1. The ASTM A300 protocol for surface treatment was employed: the steel rods were (i) pre-cleaned with degreaser, (ii) acid-cleaned with nitric acid, (iii) rinsed with distilled water in order to enrich the concentration of surface hydroxyl groups, (iv) dried under vacuum, and finally (v) the surfaces that will contact the epoxy resin were roughened with sand paper.

2. The silane coupling agent solution was prepared as follows: (i) an aqueous alcohol solution with 40ml ethanol (95% purity) and 2ml distilled water was prepared where the pH of solution was adjusted between 4.5 to 5.5 with acetic acid, (ii) a 3-mercaptopropyl trimethoxysilane (Sigma Aldrich) coupling agent was added so that the final concentration of the coupling agent was 5% and (iii) the solution was allowed to sit for 5 minutes before applying to the steel rods in order to allow hydrolysis and silane formation. The silane coupling agent has a mercaptopropyl functional group that can react with the epoxy resin and a methoxy group that will react with hydroxyl groups on steel surface.

3. The steel rods were dipped into the silane coupling agent solution, agitated gently for 10 minutes, rinsed with ethanol, and dried in vacuum oven at 110°C for 2 hours.

4. An uncured epoxy was prepared as described previously in the Material Synthesis section.

5. The longitudinal test specimen was prepared by (i) placing the annular Teflon spacer on one of the steel rods, (ii) placing approximately 0.2 ml drops of uncured epoxy resin on the surface of the rod that had an annular Teflon spacer, (iii) bringing the second rod in contact with the first rod with

9

Teflon spacer and pressing the two rods together in an special jig that facilitated axial alignment and (iv) putting the rod-spacer-epoxy assembly into a vacuum oven and using the appropriate two stage cure schedule for the epoxy resin being tested.

6. The cured longitudinal test specimen with two steel rods, Teflon spacer and cure resin was allowed to cool to room temperature in a vacuum oven to prevent moisture adsorption, the alignment jig was removed and the sample stored in a desiccator in order to prevent the adsorption of water in the epoxy layer between the two steel rods.

7. The cured test specimen was then instrumented with a minimum of three strain gages (Vishay

Micro-Measurement C2A-06-125W-350) that were attached around the circumference of thin epoxy layer.

8. It is well known that the strength of the adhesive bond between epoxy and steel, even with a coupling agent, is vulnerable to water vapor.[43-45] Consequently the circumference of the test fixture was coated with a thin layer of a butyl rubber paste (M-Coat FBT, Vishay Micro-Measurement) that was subsequently cured at room temperature for two days. Butyl rubber is a rubber with a relatively low moisture permeability.[46]

9. After curing the butyl rubber, the longitudinal test specimen was stored inside a desiccator prior to testing in the mechanical test rig.

The use of the Teflon spacer is critical, since it ensures that the epoxy layer has both uniform thickness

and a well-defined radius that is centered on the steel rods in order to ensure good axial alignment and,

consequently, a homogeneous deformation field in the longitudinal deformation experiment. The

modulus of the butyl rubber is two or more orders of magnitude less than that of the epoxy resin; thus the

thin butyl rubber layer does not contribute to the axial stress. The Teflon spacer does not adhere to the

steel and consequently also does not contribute to the axial force.

10

epoxy layer

steel

steel

Teflon spacer

strain gage

\\

\\

Figure 2: Schematic of thin epoxy layer between steel rods: thickness of epoxy layer is 350μm, diameter of steel plates is 2.54cm. Three or more strain gages (Vishay Micro-Measurement C2A-06-125W-350) are placed around the circumference. Loading direction is shown by block arrows.

For the DGENG epoxy cured with 4,4’MDA test specimen, the cure schedule was 80°C for 4hrs

followed by the second step at 130°C for 3hrs. The thickness of the cured specimen was measured by

optical microscopy, where the thickness of the epoxy layer was 350+14μm (i.e. 4% standard deviation)

and thus for an epoxy layer the ratio of thickness to diameter was 0.017. The epoxy-steel longitudinal test

specimen of DGENG-4,4’MDA with the silane coupling agent supported a stress of 45.5MPa (at Tg-

20°C) as compared to a test specimen with untreated surfaces that exhibited adhesive failure at 3.8MPa.

The specimen for the longitudinal experiment was mounted in an Instron 5567 testing system with a

thermal chamber supplied with –35oC dew point dry air to minimize moisture exposure. A flexible

coupler and screw adapter were used to accommodate alignment along the vertical axis. The effectiveness

of the longitudinal test assembly in ensuring the self-alignment was tested by mounting a solid stainless

steel rod (without the epoxy layer) on the Instron and performing the axial deformation at a constant

strain rate with six strain gages attached around the rod’s circumference. The data in Figure 3

demonstrates the excellent alignment, where the average Young’s modulus was 193GPa with a 3%

11

standard deviation, which is in good agreement with the literature value of 200GPa. In all subsequent

tests a minimum of three strain gauges are used, where the axial displacements measured by the strain

gauges were always within +4%.

0 100 200 3000

10

20

30

40

50

micro strain

engi

neer

ing

stre

ss (M

Pa)

1, 203GPa2, 197GPa3, 189GPa4, 188GPa5, 186GPa6, 195GPa

Figure 3: Alignment check for longitudinal test fixture using a solid steel rod. Outputs from six strain gages with respective moduli are shown. (Colors online)

The data in Fig. 3 demonstrate that the deformation is axially uniform (i.e. one side of the epoxy

specimen does not experience a different axial strain than the opposite side). This spatially uniform axial

deformation will result in a longitudinal deformation of the epoxy layer if the diameter of the epoxy layer

is infinite and the steel rods do not exhibit a lateral contraction. However, the epoxy near the edge will not

experience a purely longitudinal deformation. Using ABAQUS/Standard, a finite element analysis of the

test specimen was performed in order to assess the magnitude of the edge effect. Both the epoxy and the

steel materials were assumed to be elastic with the Young’s moduli (steel - 200GPa; epoxy - 2.9GPa) and

Poisson’s ratios (steel – 0.3; epoxy – 0.4).[3] The finite element analysis shows the steel parts

experience a uniaxial deformation, and the epoxy layer is under almost perfect longitudinal deformation

except at the edges, where the radial strain (which should be zero in a longitudinal deformation) is less

than 2% of the axial strain. As is shown in the Figure 4 the epoxy layer experiences homogeneous

12

deformation except for the very edge, which is negligible to the stress generated by the whole epoxy

volume.

0 2 4 6 8 10 120

0.5

1

1.5

element from the center

max

imum

prin

cipa

l str

ain

Figure 4: The maximum principal strains in the epoxy layer as determined from an ABAQUS finite element analysis with an epoxy thickness=340μm and epoxy diameter=20.32mm. In the ABAQUS figure, the color corresponds to the value of the maximum principal strain (MPS) for each element where the red, yellow, and green elements have the MPS value that is 600 times, 425 times, and 275 times greater than

that for the blue elements, respectively. The ratio of radial to axial strain ( ) is -0.018 ± 0.003 for epoxy layer and -0.3 ± 0.001 for steel disks. The plot clearly shows homogeneous deformation except for the edge element.

The signal from the strain gage shown in Fig. 2 is due to both the length change in the epoxy layer

and the length change in the steel rods to which the strain gage is attached. The initial length of the strain

gage is 3175μm and initial length of the epoxy layer is approximately 340μm, where the exact length is

13

measured for each sample by optical microscopy. The output of the strain gage, , is related to the

strains in the steel, , and in the epoxy layer, , as follows

(2)

where is the initial gage length and is the initial epoxy layer thickness. For a given stress the strain

in the steel is computed using and Young’s modulus of 200GPa for steel.[47] Using this

information, the strain in the epoxy layer, , is determined from the strain gage data, , and the

geometry of the test fixture.

4. Results

4.1. PVT behavior

PVT measurements were performed using Gnomix pressure dilatometer, where PVT data for the

DGENG-44’MDA system are shown in Figure 5 for pressures between 10 to 200MPa and temperatures

between 30 to 160°C. The slope of the 10MPa isobar changes at approximately 70°C, indicating the

volumetric Tg that is consistent with the DSC Tg of 72°C. Polymer PVT data are often fit with the

empirical Tait equation given by[41, 48]

(3)

where V0 is the specific volume at P=0atm and 0°C, and α0 is the thermal expansivity at P=0atm and B0

and B1 are two Tait parameters. The Tait Eqn. 3 was fit to the DGENG-44’MDA PVT data in Fig. 5,

14

where the Tait parameters are given in Table 2. The V(T) response at atmospheric pressure was

determined by extrapolation of the Tait Equation. The isothermal bulk modulus, ,

was calculated from the Tait Equation and is shown in Fig. 6. The PVT surfaces, Tait equation fits to the

data and the isothermal bulk modulus were also determined for EPON825 cured with a stoichiometric

amount of 33’DDS, where the results are given in the Supplemental Information.

20 50 80 110 140 1700.81

0.83

0.85

0.87

0.89

temperature (oC)

spec

ific

volu

me

(cm

3 /g)

Figure 5: Volume isotherms of DGENG-44’MDA (see text for temperature-pressure history). , □:10MPa, ○:25MPa, ＋: 50MPa, △:75MPa, ◁:100MPa, ▷:125MPa, ◇: 150MPa, ▽: 175MPa, X: 200MPa (colors online)

15

Table 2: Tait constants

V0 (cm3/g) α0 (°C-1) B0 (MPa) B1 (°C-1)

glassy 0.8429 1.652 × 10-4 473.7 1.271 × 10-3

rubbery 0.8208 5.336 × 10-4 376.3 4.190 × 10-3

20 70 120 1701

2

3

4

5

6

temperature (oC)

bulk

mod

uls

(GPa

)

Figure 6: Bulk modulus vs. temperature at atmospheric pressure for DGENG-44’MDA (Tg=72°C)

4.2. Longitudinal Modulus for High Tg Epoxy Systems

16

Longitudinal test specimens were made according to the procedure outlined in Section 3 using the

cure schedules shown in Table 1 for EPON825 cured with 33DDS and EPON825 cured with AEP, where

the Tg’s are respectively 170°C and 125°C. The longitudinal test specimens with three strain gages placed

around the circumference were deformed at a constant axial deformation rate corresponding to a strain

rate of 1.2x10-5s-1. The stress-strain behavior at 25°C is shown in Figs. 7 and 8 for EPON825-33DDS and

EPON825-AEP, respectively. There is good axial alignment with less than 4% difference between the

three strain gages until the beginning of failure. For the EPON825-AEP material, all three strain gages

had the same output until brittle failure at 0.8% strain. For the EPON825-33DDS material brittle failure

was observed at 0.5% strain, where one of strain gages measured a larger displacement starting at

approximately 0.35% strain - perhaps the consequence of formation of a small crack during deformation.

The failure for both epoxy systems is clearly cohesive, where the epoxy layers remain on both steel

surfaces as shown in Figs. 7 and 8. The stress-strain curves up to the beginning of failure appear linear,

17

0.000 0.002 0.004 0.0060

5

10

15

20

25

30

35

strain

engi

neer

ing

stre

ss (M

Pa)

site1site2site3

Figure 7: Longitudinal stress-strain curve of EPON825-33DDS at 25°C with pictures of failure surfaces.

0.000 0.002 0.004 0.006 0.0080

5

10

15

20

25

30

35

strain

engi

neer

ing

stre

ss (M

Pa)

site1site2site3

Figure 8: Longitudinal stress-strain curve of EPON825-AEP at 25°C with pictures of failure surfaces.

which enable determination of the linear elastic longitudinal moduli, M, the latter is defined as the ratio of

axial stress to axial strain where all other strains are zero, reported in Table 3.

For an isotropic linear elastic material there are well-known relationships between the various linear

elastic material properties,[49, 50] which will now be used to evaluate the accuracy of the modulus

determined in the longitudinal test. For the EPON825-33DDS system at 25°C the Young’s modulus, E,

and the Poisson’s ratio, ν, were measured using an Instron 5567 test machine in uniaxial tension; the bulk

modulus, K, was measured using a Gnomix dilatometer using the protocol discussed previously in this

Section; and, the shear modulus, G, was measured using ARES-LS2 rheometer (TA Instruments). The

experimentally determined moduli and Poisson’s ratio are given in the Table 4. Also shown in Table 4

are G, K, and M moduli calculated from the experimentally measured E and ν using the standard

formulae[49, 50]

(4)

18

Table 3: Average modulus, M, as measured by three strain gages in longitudinal deformation for EPON825-33’DDS and EPON825-AEP at 25°C

EPON825-33’DDS

site longitudinal modulus(GPa)

average(GPa)

standard dev.(GPa)

error(%)

1 6.46.3 0.2 +3.22 6.1

3 6.5

EPON825-AEP

site longitudinal modulus(GPa)

average(GPa)

standard dev.(GPa)

error(%)

1 4.14.1 0.2 +4.92 4.2

3 3.9

The calculated value of the longitudinal modulus agrees with the experimentally measured modulus

within better than 3%. The largest discrepancy between the calculated and the experimental values is

observed for the shear modulus. In case of the EPON825-AEP system only the Young’s modulus and the

Poisson’s ratio were independently measured and used to calculate the longitudinal modulus value, where

the agreement between the calculated and the experimentally measured values is within 1%.

Table 4: Elastic constants for EPON825-33DDS and EPON825-AEP at 25°C

EPON825-33DDS moduli errorcalculated experimentalYoung’s, E (GPa) - 3.10 -

Poisson’s, ν - 0.397 -shear, G (GPa) 1.11 1.04 6.7 %bulk, K (GPa) 5.02 5.10 1.6 %

longitudinal, M (GPa) 6.50 6.32 2.8 %

Epon825-AEP moduli errorcalculated experimentalYoung’s, E (GPa) - 2.55 -

Poisson’s, ν - 0.35 -longitudinal, M (GPa) 4.09 4.07 5.0 %

Experiments on the high Tg epoxy systems demonstrate that the newly-designed setup for the

longitudinal testing successfully creates longitudinal deformation conditions. Nevertheless for the stress-

strain curves in Fig. 7 and 8, yield was not observed at 25°C because brittle failure occurred first. It is

expected that ductile behavior would be observed if the test temperature was increased to just below the

glass transition temperature.[4, 8, 38] However, high temperature longitudinal experiment is not possible

using the current setup due to the increased sensitivity of the epoxy to moisture at high temperatures,

temperature limitations of the strain gages and degradation of the butyl rubber at high temperatures.[51]

A lower Tg epoxy system is required to observe the longitudinal yield. The preferred system would have a

glass transition temperature of approximately 70°C, which is high enough to obtain the glassy modulus

19

from PVT data and at the same time less than 80°C that is the upper limit of the working range of the

strain gage and butyl rubber.

4.3. Longitudinal Modulus and Yield for Lower Tg Epoxy Systems

A series of epoxy systems were synthesized and their Tg’s were measured by DSC on heating at

10°C/min. Various combinations of di-epoxides and amine curing agents, associated curing schedules and

the resulting glass transition temperatures are given in Table 5. Note that generally shorter and less

flexible chain segments between crosslinks result in the higher glass transition temperature. The epoxy

system of DGENG (neopentyl glycol diglycidyl ether) cured with 44’MDA (methylenedianiline) has the

desired glass transition temperature of 72°C.

Longitudinal stress-strain curves for DGENG-44’MDA are shown in Figure 9 at four temperatures

just below Tg. The Young’s moduli for the DGENG-44’MDA material for the same temperatures have

been measured in the uniaxial tension and the bulk moduli have been measured as described in Section

20

Table 5: Epoxy systems synthesized, curing steps and their Tg values are included. The glass transition temperature is measured by DSC on heating at the scan rate of 10°C/min.

di-epoxide curing agent Thermal profile during cure Tg

DER732(DER736)

mPDA80°C(4hrs)+130°C(3hrs)

-25.7°C (28.1°C)

44’MDA -20.4°C (31.7°C)

33’DDS 120°C(4hrs)+180°C(3hrs) -14.7°C (35.4°C)

DGEB44’MDA 80°C(4hrs)+130°C(3hrs) 57.1°C

33’DDS 120°C(4hrs)+180°C(3hrs) 64.9°C

DGENG44’MDA 80°C(4hrs)+130°C(3hrs) 71.8°C

33’DDS 120°C(4hrs)+180°C(3hrs) 78.2°C

EPON825 Versamid 140 60°C(24hrs)+150°C(24hrs) 109°C

4.1 and are given in Table 6. The longitudinal modulus calculated from E and K is also shown in Table 6,

where the calculated M is in good agreement with experimentally measured longitudinal moduli at

temperatures from Tg-5°C to Tg-20°C. This confirms that in the linear limit the longitudinal experiment

with the DGENG-44’MDA material is consistent with independent experiments. Examining the

longitudinal stress-strain curve at Tg-5°C shown in Figure 9a, the stress-strain curve displays three distinct

regimes: (i) a linear region at small strains, (ii) a second linear region with a reduced slope and (iii) a final

flat region that ends in a cohesive failure. Qualitatively similar results were obtained for DER736 cured

with 33DDS at Tg-10°C (Tg=35°C) and for an off-stoichiometric mixture of EPON825 cured with AEP

(specifically, 30% AEP by weight resulting in a molar ratio of EPON825:AEP of 3.0:0.6) at Tg-10°C

(Tg=55°C) (see Supplemental Information). At lower temperatures cohesive failure occurred in the second

linear region of the stress-strain curve. We postulate that the beginning of the flat portion of the curve

corresponds to cavitation, where the flat portion is the growth of macroscopic cracks which would occur

at a nearly constant axial stress.[52] Investigation of the failure behavior is beyond the scope of this

paper; instead, the focus of this paper is on the initial change in slope, which we postulate is longitudinal

21

Table 6: Elastic constants for DGENG-44’MDA from Tg-20°C to Tg-5°C (in GPa). M was calculated from E and K using Eqn. (4).DGENG-44’MDA

modulierror

calculated experimental

Tg-20°CYoung’s, E - 1.09 -

bulk, K - 4.81 -longitudinal, M 5.31 5.40 1.7%

Tg-15°CYoung’s, E - 0.66 -

bulk, K - 4.69 -longitudinal, M 4.99 5.06 1.4%

Tg-10°CYoung’s, E - 0.37 -

bulk, K - 4.53 -longitudinal, M 4.69 4.56 3.0%

Tg-5°CYoung’s, E - 0.09 -

bulk, K - 4.38 -longitudinal, M 4.42 4.25 4.0%

yield. The location of the yield point is determined from the intersection of two linear asymptotes as

shown in Figure 9.

0.000 0.005 0.010 0.0150

10

20

30

40

strain

engi

neer

ing

stre

ss (M

Pa)

(a) Tg-5oC

0.000 0.005 0.010 0.0150

10

20

30

40

strainen

gine

erin

g st

ress

(MPa

)

(b) Tg-10oC

0.000 0.005 0.010 0.0150

10

20

30

40

strain

engi

neer

ing

stre

ss (M

Pa)

(c) Tg-15oC

0.000 0.005 0.010 0.0150

10

20

30

40

strain

engi

neer

ing

stre

ss (M

Pa)

(d) Tg-20oC

Figure 9: Longitudinal stress-strain curves for glassy DGENG-44’MDA at various temperatures below Tg=72°C. Dashed lines are linear asymptotes and yield is indicated by the arrow.

Since there is a possibility that the observed knee on the stress-strain curve may come from some type

of damage, the following sequence of additional longitudinal tests was carried out. First, the stress-strain

response at Tg-5°C for the DENG-44’MDA shown in Fig. 9 was re-plotted in Fig. 10 as Experiment 1

(Exp 1). Then a new longitudinal DENG-44’MDA test specimen was manufactured and deformed at T g-

5°C to 13MPa, where the stress-strain response is denoted as ‘Exp 2 - 1 st pull’ in Fig. 10. 13MPa is

greater than the yield stress, but is well below the failure stress as determined in Exp 1. The stress-strain

response in ‘Exp 2 - 1st pull’ is identical with corresponding portion of the ‘Exp 1’ clearly demonstrating

22

the reproducibility of the longitudinal testing at least through the yield region. Upon reaching 13MPa the

Exp 2 longitudinal specimen assembly was released from the grips and allowed to relax for 5 mins at T g-

5°C, after which time the specimen assembly was re-gripped. A second stress-strain curve was initiated

and the response is shown as ‘Exp 2 - 2nd pull’ in Fig. 10, where the Exp 2 - 2nd pull stress-strain curve

overlaps with both the ‘Exp 2 - 1st pull curve’ and the ‘Exp 1’ curve up to 16MPa – which includes the

longitudinal yield. This clearly demonstrates that the material sustained no permanent damage during the

first pull, which included deformation through the yield point. Note that unlike the yield stress, the

ultimate failure stress in ‘Exp 2’ is different from that in ‘Exp 1’, which is expected for the cavitational

failure that is flaw sensitive.[17, 18]

0.000 0.005 0.010 0.015 0.0200

5

10

15

20

25

strain

engi

neer

ing

stre

ss (M

Pa)

Exp2, end of 1st pull

Exp2, 2nd pull

Exp1

yield

Figure 10: Loading, unloading and reloading longitudinal experiment for DGENG-44’MDA at Tg-5°C: Exp1 (dashed blue), Exp2 consisting of 1st pull (solid red), unloading (not shown), and 2nd pull (solid green). (Colors online) Longitudinal yield point is indicated by arrow.

23

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

strain

engi

neer

ing

stre

ss (M

Pa) (a)

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

60

strain

engi

neer

ing

stre

ss (M

Pa) (b)

Figure 11: Stress-strain curves for the DGENG-44’MDA epoxy system in (a) uniaxial tension and (b) uniaxial compression at temperatures from Tg-40°C to Tg-5°C (red: Tg-40°C, blue: Tg-30°C, green: Tg-20°C, purple: Tg-15°C, brown: Tg-10°C, and grey: Tg-5°C). Yield point is determined as the stress maximum (circle) or the intersection of two asymptotes when the maximum is not observed (dashed line). (Colors online)

The stress-strain behavior through yield was measured for the DGENG-44’MDA epoxy resin for

uniaxial tension and uniaxial compression for a series of temperatures and are shown in Fig. 11. In order

to compare the stress-strain behavior in different three-dimensional deformation states for a viscoelastic

material, the other factors specifically thermal history and strain rate were kept the same. [4, 14, 19] The

strain rate was fixed to 1.2x10-5s-1 for all deformations in this communication. This is a relatively slow

24

rate compared to the majority of studies in the literature; however, it is the rate at which the longitudinal

experiment could be reliably carried out. All specimens were subject to the same thermal history which

consisted of (i) heating to 102°C (i.e. Tg + 30°C) to remove the effect of any previous deformation, (ii)

quenching through exposure to room temperature for a period of 5 mins, and (iii) thermally equilibrating

for 30 mins at the test temperature prior to initiating the deformation. Finally, subsequent to manufacture

all test specimens were kept inside a desiccator prior to testing in order to eliminate any moisture

absorption. The yield point was determined as the point where the stress passed through maximum or the

intersection point of two asymptotes on the stress-strain curve when such a maximum did not exist. When

plotted as a function of temperature in Fig. 12, the yield stresses in both tension and compression show

nearly linear dependence on T-Tg, where slight deviation is observed close to Tg.

-40 -30 -20 -10 00

10

20

30

40

50

60

T-Tg (oC)

yiel

d st

ress

(GPa

)

Figure 12: The yield stresses for the DGENG-44’MDA epoxy system for uniaxial tension (squares) and uniaxial compression (circles) at temperatures from Tg-5°C to Tg-40°C.

For the purpose of comparing deformations under three different conditions, including longitudinal

tension, uniaxial tension, and uniaxial compression, all three stress-strain curves are plotted in Fig 13,

where the longitudinal modulus, M, is much greater than that in uniaxial extension or compression, which

25

is consistent with and the bulk modulus, K, being much larger than the shear modulus,

G. The stress in the longitudinal deformation continues to strongly increase after yield as compared to

uniaxial tension and compression. This can be understood from the perspective of yield phenomenon as

deformation induced mobility increase that is similar to the increase in mobility with temperature at T g,

where the post-yield behavior is controlled by the material parameters of the rubbery state. Depending on

the degree of cross-linking, the shear modulus may decrease by several orders-of-magnitude as the

material moves from the glassy to the rubbery state; in contrast, the bulk modulus only decreases by a

factor of 2 to 3 between the glass and the rubber as shown in Figure 6. Thus, it is reasonable that there is

less of a change in the slope of the stress-strain curve at yield for longitudinal deformation that is

controlled primarily by the bulk modulus versus shear dominated uniaxial deformations. The values of the

elastic moduli and yield parameters are summarized in Table 7, where the moduli for uniaxial tension and

compression (each based on three samples average) differ by less than 5%, which is consistent with the

accuracy of all other material properties reported in this communication.

0.00 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

strain

long

itudi

nal s

tres

s (M

Pa)

0

1

2

3

4

5

6

unia

xial

str

ess

(MPa

)longitudinal

compressive

tensile

26

Figure 13. Stress-strain curves for the DGENG-44’MDA epoxy system at Tg-10°C for the following deformation states; uniaxial tension (black), uniaxial compression (red), and longitudinal tension (blue). (Colors online)

Table 7: Values of elastic moduli and yield parameters for DGENG-44’MDA at Tg-10°C.

configuration E or M(GPa)

yieldstrain (%) stress (MPa)

uniaxial tension 0.38 1.0 3.4uniaxial compression 0.40 1.0 4.1longitudinal tension 4.56 0.3 13.6

4.4. Pressure-Modified von Mises Yield Criterion

As discussed in the Introduction, the original von Mises yield criterion developed for metals has been

modified to predict yield phenomena in glassy polymers, where the yield stress is assumed to have a

linear dependence on hydrostatic pressure.[7, 8, 15] The von Mises yield criterion is most naturally

expressed in terms of the octahedral shear stress, , and pressure, P, as given by

(5)

For a linear elastic material the expressions for the stress/strain tensors, octahedral shear stresses, and

hydrostatic pressures in terms of principal stresses are given in Table 8 for uniaxial extension, uniaxial

compression and longitudinal deformations. The yield stress in terms of and P is shown in Fig. 14

for the DGENG-44’MDA epoxy for the three types of deformations for four temperatures between Tg-

27

5°C to Tg-20°C. The longitudinal yield is in a different region of the space, providing data that is

much closer to the pure hydrostatic extension line, i.e. the axis. The longitudinal yield data are

consistent with the pressure-modified von Mises criterion with a linear pressure dependence and R2

correlation coefficient of 0.99. The error bars on the data in Fig. 14 are the standard deviations

determined based on three to five samples. Because of the close proximity of the longitudinal yield data

to the line, the data in Fig. 14 provide clear evidence that yield does occur in hydrostatic

extension, i.e. at negative pressures.

-40 -30 -20 -10 0 100

2

4

6

8

10

12

14

hydrostatic stress (MPa)

octa

hedr

al s

hear

str

ess

(MPa

)

28

Figure 14: The yield stress in terms of τoct vs. P for the DGENG-44MDA epoxy system at temperatures from Tg-5°C to Tg-20°C, ▲: Tg-20°C (R2=0.9965), ●: Tg-15°C (R2=0.9996), ■: Tg-10°C (R2=0.9997), ★: Tg-5°C. The lines are linear regressions. Vertical and horizontal error bars are shown. (Colors online)

Table 8: Expressions for stresses and strains in terms of principal stresses under the assumption of linear elasticity

uniaxial tension uniaxial compression longitudinal tension

stress

strain

octahedralshear stress

pressure

At any given temperature the pressure-modified von Mises criterion results in a conical yield surface

in a three dimensional space of the principal stresses (σ1, σ2, σ3). In the pressure-octahedral shear stress

coordinate space the pure hydrostatic deformation line (x-axis) and stress at τoct = 0 (x-intercept)

correspond to the main axis of the cone and the apex of the cone, while the regression line is the lateral

surface of the cone. In order to study the features of the cone-shaped yield surfaces for a wider

temperature range, additional uniaxial experiments were conducted at Tg-30°C and Tg-40°C, where the

yield stress is given in Figure 15. Longitudinal tests at these lower temperatures were not performed since

the predicted yield stress based on modified von Mises criterion (43 and 57MPa at Tg-30°C and Tg-40°C)

is larger than the adhesive failure stress (about 40MPa). The yield behavior can be described by the linear

relationship between and P as given in Eqn. 1, where the empirical coefficients and are

temperature dependent and are given in Table 9 and plotted in Fig. 16. The coefficient of internal friction

μ, i.e. the slope of the octahedral shear stress vs. pressure, is constant for temperatures below Tg-30°C,

29

which is well-known;[5, 29, 36, 53] however, μ decreases to zero as temperature approaches Tg. We

believe this is the first report of in the vicinity of Tg. Both and the apex of the cone (i.e. P at τoct = 0)

shows a linear dependence on T-Tg, where there is a slight deviation from linearity as Tg is approached.

This is the first report on the cone apex dependence on temperature, which is now reliable because the

longitudinal yield data eliminate the need for large extrapolation of uniaxial data to the τoct = 0 line.

-40 -30 -20 -10 0 10 200

5

10

15

20

25

30

hydrostatic stress (MPa)

octa

hedr

al s

hear

str

ess

(MPa

)

30

Figure 15: The yield stress in terms of τoct vs. P for the DGENG-44MDA epoxy system at temperatures from Tg-5°C to Tg-40°C ▼: Tg-40°C, ◆: Tg-30°C, ▲: Tg-20°C, ●: Tg-15°C, ■: Tg-10°C, ★: Tg-5°C. The lines represent linear regressions. Vertical and horizontal error bars are shown. (Colors online).

Table 9: μ, , and P (τoct = 0) at temperatures indicated between Tg-5°C and Tg-40°C,

temperature Tg-40°C Tg-30°C Tg-20°C Tg-15°C Tg-10°C Tg-5°C

μ (slope) 0.31 0.31 0.27 0.19 0.12 0.05

21.0 15.2 9.6 4.6 1.8 0.4

P (τoct = 0) -66.8 -49.0 -35.8 -23.5 -14.8 -9.7

-40 -30 -20 -10 00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

, s

lope

T-Tg (oC)

(a)

-40 -30 -20 -10 00

5

10

15

20

25

oct

o

T-Tg (oC)

(b)

-40 -30 -20 -10 0

-75

-60

-45

-30

-15

0

P (

oct =

0)

Figure 16: Temperature dependence of (a) the coefficient of friction μ, (b) the apex of the yield surface

given by P (τoct = 0) and model parameter for DGENG-44’MDA from Tg-40°C to Tg-5°C. In (b) : circles and P (τoct = 0): diamonds

5. Discussion

31

In this communication, we report the first observation of glassy polymer yield in a longitudinal

deformation. A specially designed experimental set-up was developed to create the dilatationally

dominated longitudinal deformation that is nearly triaxial extension. A thin epoxy layer was cured

between a pair of steel rods and the epoxy/rod assembly was pulled axially in a tensile testing rig, where

the rods were connected to the Instron grips via self-aligning adapters to insure a uniform longitudinal

deformation field. The yield phenomena were studied in the region just below Tg for a DGENG epoxy

cured with 44’MDA that had a Tg=72°C as determined by DSC. Of particular significance is that the

volume change at yield for the longitudinal deformation is one or two orders of magnitude higher than in

the uniaxial extension. Specifically, if the material is assumed to be linear elastic up to yield, then the

relative volume change at yield is given by

(6)

where the axial strain is evaluated at the yield. Using the linear elastic material properties for DGENG-

44’MDA epoxy, the relative volume change at yield is shown in Table 10. At Tg-5°C the longitudinal

volume change at yield is more than two orders-of-magnitude greater than the volume change in uniaxial

extension at yield, where at lower temperatures the difference in volume change at yield between

longitudinal vs. uniaxial extension is not as large but still significant.

32

Table 10: Relative volume change at yield for longitudinal and uniaxial extension for DGENG-44’MDA

temperature Tg-5°C Tg-10°C Tg-15°C Tg-20°C

longitudinal tension

yield strain 0.002 0.003 0.004 0.006

ΔV/V0 2.0E-03 3.0E-03 4.0E-03 6.0E-03

uniaxial tension

yield strain 0.009 0.010 0.013 0.015

ΔV/V0 6.3E-06 2.1E-04 4.8E-04 9.8E-04

ΔV/V0 ratio 316 15 8 6

DGENG-44’MDA yield data in the dilatationally dominated longitudinal deformation were combined

with the traditional measurements of yield in uniaxial tension and uniaxial compression tests where the

deviatoric contribution dominates. The combination of longitudinal, uniaxial extension and uniaxial

compression yield data obeys the empirical pressure-modified von Mises yield criterion, where the

longitudinal yield data significantly expand the pressure axis as compared to that from just yield in

uniaxial extension and compression. Using just the uniaxial tests without any longitudinal data, the slope

μ of the pressure-modified von Mises criterion for temperatures below Tg-30°C approaches a constant

value of 0.31 that is higher than the literature values of 0.13 to 0.24 for other epoxy systems (see Table

11). decreases linearly with temperature as the test temperature approaches Tg (Fig. 16a). The values

in this temperature range are determined with greater confidence on the basis of three experimental

points, one of which is due to the longitudinal test. We believe that this is the first report of the decrease

in μ in the vicinity of Tg. Examining the data in Fig. 16b, both and the apex of the cone (i.e. P at τoct =

0) show linear dependence on T-Tg, where there is a slight deviation from linearity as Tg is approached.

33

Because the longitudinal deformation is predominantly hydrostatic, for the first time it is possible to

accurately determine the apex of the yield surface cone (i.e. P at τoct = 0) in the principal stress space vs.

using the deviatorically dominated measurements like uniaxial extension and compression that require a

large extrapolation. This is the first report on the cone apex dependence on temperature.

Table 11: Literature reports on μ and its dependence on temperature for a variety of polymers

system Temp. μ (slope) Ref.

Epoxy(EPON828 / Shell’s curing agent D) RT 0.18 Sultan

and McGarry[1]Epoxy

(EPON825 / 1,3-phenylenediamine)Tg-130°C ~ Tg-40°C 0.23 Lesser

and Calzia[8]Epoxy

(EPON825 / ethylenediamine)Tg-120°C ~ Tg-50°C 0.17 Kody

and Lesser[11]Epoxy

(DER332 / diethyenetriamine) RT 0.24 Asp and Berglund[13]

Epoxy(DER332 / Jeffamine T403) RT 0.13 Asp

and Berglund[13]Epoxy

(Ciba MY750 / DY040) RT 0.19 Bowden and Jukes[15]

PMMA Tg-50°C ~ Tg-30°C 0.15 Sternstein

and Ongchin[20]PC

(Makrolon Bayer)Tg-200°C

~ Tg-30°C 0.075 Bauwens-Crowet and Bauwens[6]

PC(Makrolon 2800)

Tg-125°C ~ Tg-80°C 0.051 Duckett

and Ward[22]

It is well established that the glass-to-rubber transition in amorphous polymers is due to a significant

increase in the molecular mobility as the temperature is increased.[4, 54] It has also been proposed that

deformation in glassy polymers can induce a similar increase in mobility,[54, 55] resulting in yield with a

dramatic change in the glassy tangent modulus. Consistent with the standard elasticity relationship M = K

+ 4G/3 and the data in Table 6, the longitudinal modulus M is an order-of-magnitude larger than both the

Young’s modulus, E, and the shear modulus, G, at Tg-15°C, where the difference increases to two orders-

of-magnitude at Tg-5°C. The fact that M is primarily controlled by the bulk modulus, K, is clearly shown

in Fig. 17, where the experimentally measured stress in the longitudinal experiment is compared with the

34

hydrostatic contribution determined from K as determined from the PVT data in Figure 6. In Fig. 17 the

initial linear portion of the stress-strain curve before the yield point is well predicted by just the bulk

modulus contribution at Tg-5°C and Tg-10°C. After the yield point the slope of the stress-strain curve

changes. Assuming that K also dominates post-yield and using the values of K in the rubbery state

determined from the PVT data shown in Fig. 6, the slope of the post-yield longitudinal stress-strain curve

at Tg-5°C has a bulk modulus associated with Tg+110°C. Similarly, the slope of the post-yield

longitudinal stress-strain curve at Tg-10°C has a bulk modulus associated with Tg+85°C. The longitudinal

stress-strain curves indicate a pre-yield glassy material and a post-yield rubbery material, consistent with

the picture of a yield being the result of deformation induced change in mobility similar to that observed

with temperature in the glass transition region, where the post-yield response is for a material well above

Tg.

0.000 0.003 0.006 0.009 0.0120

5

10

15

20

25

strain

engi

neer

ing

stre

ss (M

Pa) K, Tg-5oC

K, Tg+110oC

(a)

Figure 17: Longitudinal test for DGENG-44’MDA at (a) Tg-5°C and (b) Tg-10°C. Dashed lines are predictions for hydrostatic stress based on the values of bulk modulus at temperatures indicated.

Yield experiments are often performed in uniaxial compression vs. uniaxial tension when studying

large deformation, because uniaxial compression experiments are less sensitive to material flaws and

gripping is easier. In this paper the longitudinal experiment was carried out in tension instead of

35

0.000 0.003 0.006 0.009 0.0120

10

20

30

40

strain

engi

neer

ing

stre

ss (M

Pa)

K, Tg-10oC

K, Tg+85oC

(b)

compression for two reasons: (i) longitudinal extension allows for the self-alignment of the specimen

assembly and (ii) longitudinal yield can only be attained in extension for the DGENG-44’MDA epoxy for

temperatures from Tg-40°C to Tg-5°C. It is instructive to estimate the value of the compressive

longitudinal yield stress on the basis of the pressure modified von Mises criterion. For the longitudinal

extension the octahedral shear stress and hydrostatic pressure expressed as functions of the axial stress are

(7)

Combining these equations one obtains

(8)

In the τoct -P plane Eqn. (8) describes a straight line with the negative slope passing through the origin as

shown in Fig. 18 for the DGENG-44’MDA epoxy. The pressure-modified von Mises yield criterion line

is also shown in Figure 18, where it has a positive slope. The linear elastic longitudinal response in

extension (i.e. Eqn. (8)) will always intersect the von Mises yield criterion line (i.e. Eqn. (1)) in the P < 0

half-plane; thus, yield will always be observed in longitudinal extension as long as a failure event does

not intervene first. For the longitudinal compression the octahedral shear stress and hydrostatic pressure

are given by

(9)

and combining the expressions in Eqn. (9)

36

(10)

which has a positive slope as shown in Fig. 18 for the DGENE-44’MDA epoxy. For compression the

linear elastic response will only intersects the yield criterion line in P > 0 half-plane if

(11)

For DGENG-44MDA the values of μ, G/K and are given in Table 12, where compression

yield is not expected because the criterion in Eqn. 11 is not met. In case of uniaxial tension and

compression, the τoct vs. P dependence is given as

(12)

(13)

which is also shown graphically in Figure 18 for DGENG-44MDA at Tg-10°C. Since is larger than μ

for the whole temperature range, there is always an intersection with the von Mises yield criterion line for

both uniaxial extension and compression. In summary, if a pressure dependent von Mises criterion

describes yield, yield will occur in uniaxial extension, uniaxial compression and longitudinal extension if

brittle and/or adhesive failure does not occur first; however, yield in longitudinal compression can only

occur if the criterion given in Eqn. (11) is satisfied, where very large compressive stresses may be

required.

37

Table 12: Experimental results for μ, G/K and for DGENG-44MDA at specified temperatures

temperature Tg-40°C Tg-30°C Tg-20°C Tg-15°C Tg-10°C Tg-5°C

μ (slope) 0.31 0.31 0.27 0.19 0.12 0.05

G/K 0.116 0.090 0.070 0.038 0.014 0.003

0.110 0.085 0.066 0.036 0.013 0.003

-15 -10 -5 0 5 10 150.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

hydrostatic stress (MPa)

octa

hedr

al s

hear

str

ess

(MPa

)

yield criterion

uniaxial

longitudinal

Yield in glassy polymers is just one aspect of the overall nonlinear viscoelastic behavior, where the

yield stress in uniaxial tension and compression depends upon thermal history, e.g. both yield stress and

post-yield stress-softening increase with sub-Tg physical aging.[1, 19] Thus the observation of

longitudinal yield reported in this paper raises the question: does the yield stress under the dilatation

dominated conditions such as the longitudinal deformation exhibit similar dependence on physical aging

38

Figure 18: Yield prediction for DGENG-44MDA at Tg-10°C: pressure-modified von Mises yield criterion (blue), uniaxial deformation (black), and longitudinal deformation (red). (Colors online)

as that observed in uniaxial experiments? Another experiment of interest is the nonlinear stress relaxation

of a glassy material before or after yield, where the initial part of relaxation response has been used to

investigate changes in the deformation induced mobility.[55] Measurement of the nonlinear viscoelastic

relaxation in uniaxial tension, uniaxial compression and longitudinal extension may help elucidate the

difference between dilatational and deviatoric contribution to the deformation induced mobility, which is

a key component of the current constitutive equations that have been proposed for glassy polymers. Thus,

the longitudinal experiments developed in this paper may provide important information concerning both

the underlying physics of nonlinear viscoelastic relaxation in glassy polymers and the

development/validation of mechanistic based constitutive models for these materials.

6. Acknowledgements

This work was supported by a grant with the Boeing Company.

7. References

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2401.

Supplemental Information

PVT Behavior of EPON825 cured with 33’DDS

The PVT behavior of EPON825 cured with 33’DDS was determined using the procedures

discussed in the main text. The PVT data are shown in Fig. A.1, where the rubber and glass PVT

data were fit independently to the empirical Tait equation (i.e. Eqn. (3)) with parameters given in

Table A.1. The temperature dependent bulk modulus at atmospheric pressure was determined

from the Tait equation as shown in Fig. A.2. The slope of volume-temperature curve at 10MPa

has a discontinuity at approximately 165oC and the bulk modulus exhibits a significant deviation

from linearity at 170oC, where both these temperatures are consistent with the Tg of 170oC as

determined by DSC.

50 100 150 200 2500.74

0.76

0.78

0.80

0.82

temperature (oC)

spec

ific

volu

me

(cm

3 /g)

41

Figure A.1: Volume isotherms of EPON825-33’DDS; for each temperature volume measurements are taken at the following pressure: □:10MPa, ○:25MPa, ＋: 50MPa, △:75MPa, ◁:100MPa, ▷:125MPa, ◇: 150MPa, ▽: 175MPa, X: 200MPa

50 100 150 200 2501

2

3

4

5

6

temperature (oC)

bulk

mod

uls

(GPa

)

Longitudinal Stress-Strain Curve for Other Epoxy Systems

Longitudinal stress-strain curves for DER736-33’DDS at Tg-10°C (Tg=35°C) and off-stoichiometric

mixture of EPON825-AEP (specifically, 30% AEP by weight) at Tg-10°C (Tg=55°C) were measured and

displayed in Figs. B.1 and B.2, respectively. As is seen in DGENG-44’MDA at T g-5°C in the main text,

qualitatively similar results were observed, where both curves displayed three distinct regimes. Initial

change in slope indicating longitudinal yield is determined from the intersection of two linear asymptotes

as shown in dashed lines. Values of elastic moduli and yield parameters are summarized in Table B.1.

42

Figure A.2: Bulk modulus vs. temperature at atmospheric pressure for EPON825-33’DDS (DSC Tg=170°C)

Table A.1: Tait constants for EPON825-33’DDS

V0 (cm3/g) α0 (°C-1) B0 (MPa) B1 (°C-1)

glassy 0.7688 1.554 × 10-4 482.9 2.158 × 10-3

rubbery 0.7212 5.526 × 10-4 551.4 4.976 × 10-3

0.00 0.01 0.02 0.03 0.040

5

10

15

20

25

30

strain

engi

neer

ing

stre

ss (M

Pa)

site1site2site3

0.00 0.01 0.02 0.030

10

20

30

40

50

strain

engi

neer

ing

stre

ss (M

Pa)

site1site2site3

43

Figure B.1: Longitudinal stress-strain curves for glassy DER736-33’DDS at Tg-10°C (Tg=35°C). Dashed lines are linear asymptotes, where intersection indicates yield.

Figure B.2: Longitudinal stress-strain curves for an off-stoichiometric mixture of EPON825-AEP (specifically, 30% AEP by weight) at Tg-10°C (Tg=55°C). Dashed lines are linear asymptotes, where intersection indicates yield.

Table B.1: Values of elastic modulus and yield parameters

configuration M(GPa)

yieldstrain (%) stress (MPa)

DER736-33’DDS 4.02 0.2 9.1EPON825-AEP 30% 4.58 0.2 11.8

44