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Stress-strain curves for solid polymers.


Although much effort (1, 2) has been expended to introduce nonlinearity in the linear theory of viscoelasticity, nonlinear behavior at finite strains is less well understood because of the many complexities peculiar to polymer solids. Many interesting and excellent experimental data on yielding and fracture for both crystalline and amorphous solid polymers have been published (3-6), and some qualitative explanations of plastic deformation have been proposed (7-9). But they seem to be far from a complete description of plastic deformation behavior. Even the definition of yielding still seems vague. Furthermore, the relationship between stress and strain in inelastic strains has been little investigated and is therefore less well understood.

For metals, on the physical basis of both the plastic potential theory and the dislocation theory, stress-strain behavior under various strain paths has been successfully described quantitatively and qualitatively. For polymers, the constitutive law proposed for metals will not be applied without modification because of the difference in the deformation mechanism in polymers and metals.

Recently, Parks and Boyce (10) proposed a constitutive model for amorphous polymers and applied it successfully to the hydrostatic extrusion of poly(methyl methacrylate) (PMMA). Vest, Amoedo and Lee (11) proposed a mechanical model consisting of four elements and described the stress-strain behavior at constant strain rate tension. Our previous papers (12-16) showed that a constitutive equation based on an overstress model proposed by Krempl (17, 18) was appropriate for the description of the stress-strain behavior without strain reversal, especially for crystalline polymers of polyethylene (PE) and polypropylene (PP). But the inelastic deformation mechanism may be different in amorphous and crystalline polymers.

In this paper, the stress-strain curves for amorphous polymers and for crystalline polymers are compared with the calculated results of the overstress theory mentioned above. And the new facts required for considering the constitutive law are pointed out for the strain paths with strain reversal.


The material used was extruded rods of a crystalline polymer of polyoxymethylene (POM) and amorphous polymers of polycarbonate (PC) and PMMA. The degree of anisotropy for the annealed rods was found to be very small. Solid cylindrical specimens of 12 mm diameter and 20 mm height and hollow cylindrical specimens with outer and inner diameters of 16 and 12 mm, respectively, and gauge length 21 mm were machined from them. After machining, they were again annealed to relieve the residual stresses resulting from machining. The former specimens were used for compression, and the latter ones were used for combined tension torsion.

The stress was calculated based on the assumption that the material is incompressible and the cylinder is thin walled. For combined tension-torsion, the tensile stress [Sigma] and the shear stress [Tau] were calculated from

[Sigma] = F(l + [[Epsilon].sub.n])/[A.sub.0] (1)

[Tau] = T[(l + [[Epsilon].sub.n]).sup.3/2]/([r.sub.0] [A.sub.0]) (2)

and the tensile strain [Epsilon] and the shear strain [Gamma] were computed from

[Epsilon] = ln(l + [[Epsilon].sub.n]) (3)

[Gamma] = ([r.sub.0]/[l.sub.0])[Theta][(l + [[Epsilon].sub.n]).sup.-3/2] (4)

where [[Epsilon].sub.n] = (l - [l.sub.0])/[l.sub.0] is the nominal strain, l and [l.sub.0] are the current and the original gauge length, F is the applied force, T is the torque, [r.sub.0] is the original mean radius of the hollow cylinder, [A.sub.0] is the original cross sectional area, and [Theta] is the twist angle over the gauge length.

An equivalent tensile strain [[Epsilon].sub.eq], its rate [Mathematical Expression Omitted], and an equivalent tensile stress [[Sigma].sub.eq] defined by

[[Epsilon].sub.eq] = [([[Epsilon].sup.2] + [[Gamma].sup.2]/3).sup.1/2] (5)

[Mathematical Expression Omitted]

[[Sigma].sub.eq] = [([[Sigma].sup.2] + 3[[Tau].sup.2]).sup.1/2] (7)

were used for comparing the experimental data at combined tension-torsion on the same diagram.

In order to measure the plastic dilatation due to compression, home-made equipment was constructed, the details of which are described elsewhere (19). Uniaxial compression tests were performed at a constant strain rate of 0.75 x [10.sup.-3]/s at 21 [+ or -] 1 [degree] C.

For the compression tests, a hydraulic servocontrolled tension-compression testing machine (Dynamic servo; Saginomiya Co., Japan) was used, and for the combined tension-torsion, a home-made testing machine was used. For some of the combined tension-torsion tests, the current strain for both tension and torsion were measured by a biaxial strain meter developed by our laboratory (20). The method of the strain measurements for the other tests is described elsewhere (13). The effect of hydrostatic pressure on the stress-strain curves was also investigated for POM using a home-made torsional machine with a high pressure chamber of 200 MPa (21). All these experiments were conducted at 25 [degrees] C.


Previous papers showed that an overstress theory proposed by Krempl, which is a slight modification of the three-element model of the linear viscoelasticity, describes the stress-strain responses in crystalline polymers such as PE, PP, and POM (12-15). The governing constitutive equation based on the assumption that the material is incompressible is written by

[Mathematical Expression Omitted]

where [] is the deviatlic stress tensor and [] is the deviatlic strain tensor, [G.sub.1] is the instantaneous shear modulus.

The multiaxial form of the constitutive equation (Eq 8) under combined tension-torsion may be described by

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

K[[Phi]] = [k.sub.0]exp{[k.sub.1][Zeta]} (11)

[k.sub.1] = [P.sub.0]exp{- [P.sub.1][(2[Theta]).sup.[P.sub.3]]} + [P.sub.2] (12)

[Phi] = [(3[[Epsilon].sup.2] + 4[[Gamma].sup.2]).sup.1/2]/2 (13)

[Zeta] = [{3[(2[Sigma]/3 - [Epsilon]g[[Theta]]/[Theta]).sup.2] + 4[([Tau] - [Gamma]g[[Theta]]/[Theta]).sup.2]}.sup.1/2]/2 (14)

where [Sigma] and [Tau] are the tensile stress and the shear stress; [Epsilon] and [Gamma] are the tensile strain and the shear strain; g[[Phi]], called the equilibrium stress-strain curve (g curve), is the stress obtained at a strain of [Phi] at infinitely slow strain rate; [Phi] is the equivalent shear strain; and [Zeta] is the equivalent overstress.

The g curve peculiar to each polymer used is determined as a curve drawn smoothly through the stresses reached after 24 h at the relaxation tests at different strains.

For the amorphous polymers exhibiting the distinct yield point, the following function is adopted using [Lambda] = exp(2[Phi]/3)

g[[Phi]] = [G.sub.0](2[Phi])exp{- [(2[Phi]).sup.q]/Q} + [[Beta].sub.1] ([[Lambda].sup.2] - 1/[Lambda]) + [[Beta].sub.2] ([Lambda] - 1/[[Lambda].sup.2] (15)

where the curve consists of the yield drop component (the first term) and the rubber elasticity component (the second terms), and [[Beta].sub.1], [[Beta].sub.2], q, and Q are the constants.

For POM, the g function is well described by

g[[Phi]] = ([g.sub.m]/f){tanh(b2[Phi]/[[Phi].sub.m]) - [(sech [b.sub.0]).sup.2](b2[Phi]/[[Phi].sub.m])}

f = b.sub.0 [(tanh [b.sub.0]).sup.2] + tanh [b.sub.0] - [b.sub.0]

b = [b.sub.1] + [b.sub.2] - [b.sub.2]exp(- [b.sub.3]2[Phi])

[b.sub.1] = f[G.sub.0][[Phi].sub.m]/[[(tanh [b.sub.0]).sup.2][[Sigma].sub.m]]

[b.sub.2] = 2[b.sub.0] - [b.sub.1] (16)

where [g.sub.m] is the maximum stress at a shear strain of [[Phi].sub.m] in the g curve and the constants [b.sub.0] and [b.sub.3] are determined to fit the experimental g data (12, 13).

Almost all of the many experimental constants are determined easily from the experimental data measured in relaxation tests. The other constant [G.sub.1] is chosen so that the equation may fit the experimental stress-strain curve at a high strain rate. When the stress-strain curve is fairly sensitive to pressure, the constants must be pressure dependent. In this paper, for POM, the pressure dependence is considered in the constants of [P.sub.0], [P.sub.2], [P.sub.3] and [g.sub.m], as shown in Table 1, where p is the hydrostatic stress component. The pressure dependent parameters are determined by the same method described in our previous papers (12-16). The constants used for the numerical calculations are listed in Table 1. In this paper, the effect of pressure is considered by making the K-function pressure dependent. This may mean that pressure dependence is equivalent to rate dependence. For polymers, it is known that there exists a role of equivalence among time, temperature, and pressure. In this viewpoint, the pressure dependence of the K-function may be considered reasonable. But this assumption may not be necessarily valid. Further work will be needed to construct a pressure dependent constitutive law on the basis of the many experiments.

In addition, an equation for the dilatational stress-strain relationship is required. However, according to the experimental measurements of the volume change due to the compression tests, the dilatation is considerably smaller than the applied compressive strain. In this paper, therefore, the materials used are assumed to be incompressible. The effect of the hydrostatic pressure on the constitutive equation should be included into the constants of [G.sub.1] and [G.sub.0] and into the equations of g[[Phi]] and [k.sub.1]. In this paper, the pressure effect was neglected for the amorphous polymers.

The computational results are compared with the experimental curves.


Dilatation Due to Compression

Typical volumetric strain ([[Epsilon].sub.v])-applied strain ([Epsilon]) curves of the polymers used are shown in Fig. 1, which also shows the stress-strain curves at the top. For the crystalline polymers of POM, PE, and PP, the stress-strain curves have no distinctive yield peak. On the other hand, for the amorphous polymers of PMMA, PC, and poly(vinylchloride) (PVC), the stress decreases with an increase in the applied strain after exceeding its peak. In the initial stage (the elastic region), the volumetric strain elastically decreases with a decrease in the compressive strain [Epsilon]. For the amorphous polymers, the volumetric strain ceases to decrease after the yield peak and is held constant in spite of an increase in the inelastic deformation. For the crystalline polymers of PE, PP, and POM, the volume strain turns to an increase near the yield peak and continues to increase with an increase in the inelastic strain. This may indicate the existence of at curious phenomenon that the dilatation is caused by the inelastic deformation, even if the first strain invariant is negative. This may be important in order to construct the constitutive law in solid polymers. However, the volume expansion is considerably smaller than the applied strain, and hence, the assumption that the material is incompressible may be appropriate.
Table 1. The Constants Used for the Calculations.

 (a) PC and PMMA


[G.sub.0] (MPa) 500 800
[G.sub.1] (MPa) 1610 1960
q 2.35 1.15
Q x [10.sup.3] 5.8 47.6
[[Beta].sub.1] (MPa) 26 8.66
[[Beta].sub.2] (MPa) 26 8.66
[K.sub.0] (s) [10.sup.4.45] [10.sup.4.8]
[P.sub.0] (1/MPa) 13.86 6.93
[P.sub.1] 8.69 19.1
[P.sub.2] (1/MPa) 0.476 0.196
[P.sub.3] 0.345 0.55

(b) POM

[G.sub.0] (MPa) 1078
[G.sub.1] (MPa) 1078
[[Phi].sub.m] 0.35
[b.sub.0] 3.0
[b.sub.3] 800
[K.sub.0] (s) [10.sup.4.9]
[g.sub.m] (MPa) 25.5 + 0.025p
[P.sub.0] (1/MPa) 6.63 - 1.5 x [10.sup.-2]p
[P.sub.2] (1/MPa) 0.306 - [10.sup.-3]p
[P.sub.3] 0.33 + 2.04 x [10.sup.-4]p
[P.sub.1] 8.5

Constant Strain Rate Loading

The effect of strain rate on the stress-strain curve in compression is shown in Fig. 2, where the solid curves are drawn based on the overstress theory described above. The cylindrical specimen did not deform into a barrel type up to a strain of 0.1 in this compression test. The stresses were calculated on the basis of the assumption that the specimen held the shape of a straight cylinder during the test. For the amorphous polymers used here, the yield drop is distinctive, which is different from the behavior of the semicrystalline polymers of PE, PP, and POM. The curves marked g are the g-curves at which relaxation terminates. For PMMA, the g curve is considerably lower than the curves obtained at constant strain rates. This shows that PMMA is strongly affected by the strain rate. The calculated results denoted by the solid curves are in good agreement with the experimental curves.

The torsional stress-strain curves for POM under pressures of 0.1 to 150 MPa are shown in Fig. 3, where the solid curves are theoretical, using the pressure dependent parameters listed in Table 1. The solid curves are in good agreement with the experimental data.

Figure 4 shows the stress-strain curves of POM and PC under tension, compression, torsion, and combined tension-torsion at an equivalent strain rate of [[Epsilon].sub.eq] = 1.47 x [10.sup.-3]/s in a [[Sigma].sub.eq] - [[Epsilon].sub.eq] diagram. The solid curves are theoretical. In the calculations for POM, the pressure dependence is considered. For combined tension-torsion, the strain path is selected as [Epsilon] = [Gamma]/[square root of 3]. For POM, the magnitude of [[Sigma].sub.eq] decreases as the first stress invariant (= [Sigma]/3) increases. But in PC, then [[Sigma].sub.eq] - [[Epsilon].sub.e1] curves are nearly the same in spite of the loading condition. This difference may be attributed to the degree of the pressure sensitivity of the polymer.

The biaxial constitutive equation (Eq 8) becomes a good approximation for combined tension-torsion as well as for uniaxial loading.


Figure 5 shows the compressive creep behavior for POM and PMMA. After compression was executed to a predetermined stress level at a constant stress rate of 5 MPa/[mm.sup.2]/s, the compressive stress was kept constant. The calculated results shown by the solid curves coincide with the experimental trends.

For the uniaxial creep, Eq 8 reduces to

[Mathematical Expression Omitted]

where [E.sub.1] = (3/2)[G.sub.1] and [g.sub.1][[Epsilon]] = 3g[3[Epsilon]/2]. Now consider the case where the g curve has a peak value [[Sigma].sub.m] (= 3[g.sub.m]) at a strain of [[Epsilon].sub.m] (= 2[[Phi].sub.m]/3) in the light of a schematic illustration of Fig. 6. In the case of the creep test where [Sigma] [less than or equal to] [[Sigma].sub.m], the creep rate [Epsilon] decreases with an increase in testing time, and finally, the creep terminates since an overstress [Sigma] - [g.sub.1][[Epsilon]] continues to decrease as the creep progresses. (See curve 1 in the Figure.) When the creep test is carried out at [Sigma] [greater than or equal to] [[Sigma].sub.m], the creep rate continues to decrease until the strain increases up to [[Epsilon].sub.m], where the overstress [Sigma] - [g.sub.1][[Epsilon]] is minimum, as expected from Eq 17 (see curve 2). But when the creep progresses and the strain exceeds [[Epsilon].sub.m], the overstress [Sigma] - [g.sub.1][[Epsilon]] begins to increase again with an increase in [Epsilon], and therefore, the creep rate increases continuously. At this stage, the creep strain increases in an accelerated fashion, and the creep strain-time curve becomes convex upwards.

We pay attention to the creep behavior of PMMA at a stress of 70 MPa, the value of which is higher than the peak stress [[Sigma].sub.m] of the g curve. The computational result explains well the experimental curve consisting of the primary stage where the creep rate decreases with an increase in time and the final stage where the creep strain increases in an accelerated fashion.

Comparison between the overstress theory and the experimental data may show that the theory describes the viscoelastic-plastic stress-strain behavior of POM and PMMA well, provided that the current strain is not below the previous strain.

Next, we consider the stress responses with strain reversal.

Strain Reversal

Figure 7 denotes the stress-strain behavior after strain reversal. After solid cylindrical specimens were compressed at a constant strain rate of [[Epsilon].sub.0] = -9 x [10.sup.-4]/s up to [[Epsilon].sub.r] = -0.1 for POM and [[Epsilon].sub.r] = -0.085 for PMMA, unloading was executed at various strain rates from -[[Epsilon].sub.0]/100 to -100[[Epsilon].sub.0]. It is interesting to note for both polymers that 1) the shape of the curves after strain reversal is strongly dependent on the strain rate, and 2) the higher the strain rate, the steeper the tangential slope just after strain reversal. The strain rate dependence of the curve after strain reversal is more complex for PMMA than that for POM.

Equation 8 is rewritten under uniaxial loading as (19)

[(d[Sigma]/d[Epsilon]).sup.+] = [E.sub.1][{1 + [[(d[Sigma]/d[Epsilon]).sup.-]/[E.sub.1] - 1]/a} (18)

where [(d[Sigma]/d[Epsilon]).sup.-] and [(d[Sigma]/d[Epsilon]).sup.+] are the tangential slopes before and after strain reversal, a = [[Epsilon].sup.+]/[[Epsilon].sup.-] is the ratio of the strain rate after strain reversal to that before strain reversal, and [E.sub.1] = 3[G.sub.1]. When [[Epsilon].sub.r] is sufficiently large, [(d[Sigma]/d[Epsilon]).sup.-]/[E.sub.1] [much less than] 1 and the equation reduces to

[(d[Sigma]/d[Epsilon]).sup.+] = [E.sub.1]{1 - 1/a} (19)

This prediction that

[(d[Sigma]/d[Epsilon]).sup.+] = 2[E.sub.1] at a = -1

[(d[Sigma]/d[Epsilon]).sup.+] = [E.sub.1] at a = -[infinity]

[(d[Sigma]/d[Epsilon]).sup.+] = [infinity] at a = 0 (20)

is in good agreement with the experimental trend just after strain reversal. But the equation is not always suitable for the description of the curve far away from [[Epsilon].sub.r].

Next, relaxation and creep tests after strain reversal were performed. The specimen was compressed up to a reversal point of [[Epsilon].sub.r] = -0.1 for POM and -0.085 for PMMA at a strain rate of 9 x [10.sup.-4]/s, subsequent unloading was executed at the same strain rate up to a predetermined strain, and the strain was kept constant (the relaxation started). The variation of stress as a function of elapsed time is shown in Fig. 8, where the result for relaxation test without strain reversal is also represented. In the case without strain reversal, the absolute value of stress continues to decrease with increasing time and tends to gradually approach a stable value. In contrast, when the magnitude of strain reversal, [Delta][Epsilon], is large, it continues to increase with increasing time until it reaches an asymptotic value. (See the marks ?? for PMMA in the Figure.) But in the case where [Delta][Epsilon] is intermediate, a curious behavior is observed, in that the absolute value of stress at first increases for a while and then decreases continuously with time until it approaches a stable value asymptotically. (See the marks ??, ?? for PMMA and ??, ?? for POM in the Figure.)

The creep behavior after strain reversal is described in Fig. 9. The specimen was compressed at a strain rate of -2.5 MPa/s for PMMA and -5 MPa/sec for POM up to a stress of -75 MPa for PMMA and -95 MPa for POM, unloading was subsequently executed at the same stress rate up to a predetermined stress, and the stress was kept constant (the creep started). As reported under Dilatation Due to Compression, in the creep test without strain reversal, the creep strain increases with increasing time for all stresses tested. But for the creep after strain reversal as well as for the relaxation, a curious behavior is observed, as shown by the marks ?? for PMMA and ?? for POM in Fig. 9. When the magnitude of strain reversal [Delta][Epsilon] is large, the strain decreases with increasing time. For example, after unloading, if the specimen is left in a stress-free state as it is, the strain recovers continuously with an increase in the elapsed time and the permanent strain remains. When [Delta][Epsilon] is small, the strain tends to increase with time. In the case where [Delta][Epsilon] is intermediate, the strain increases after it decreases. This indicates that the sign of the strain rate may change even during a creep test at a constant stress. This phenomenon is similar to that of the relaxation after strain reversal.

These phenomena observed at both the relaxation and the creep after strain reversal, which seem anomalous, may not have been pointed out previously. We shall call these phenomena anomalous behaviors associated with strain reversal. This anomalous behavior may probably be attributed to strong time-sensitive characteristics of polymers. The temperature alteration during the test may not be a principal reason for this behavior, since it is only less than 1 [degree] C and hence is too small to cause the anomalous stress variation through thermal expansion. Unfortunately, the reason for this anomalous stress variation is unclear.

Such behavior may be very different from that of metals. This may show that it is difficult to construct an all-around constitutive equation taking into account the complex behavior after strain reversal as well as the stress response at monotonous loading.


In order to investigate the constitutive equation for solid polymers, stress-strain behavior was observed using an amorphous polymer of poly(methyl methacrylate) (PMMA) and a crystalline polymer of polyoxymethylene (POM) for some strain paths such as uniaxial tension, torsion and combined tension-torsion in a strain controlled test. It is shown that 1) the volumetric expansion due to inelastic deformation occurs depending on the material even if the current value of the first stress invariant is negative, 2) an overstress theory proposed by Krempl fairly well describes the stress-strain curves of both POM and PMMA at the strain paths without strain reversal, and 3) in the stress relaxation or the creep executed after strain reversal, an unexpected behavior in which the stress rate or the strain rate may change its sign depending on the test condition is observed.


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21. M. Kitagawa, J. Qiu, K. Nishida, and T. Yoneyama, J. Mater. Sci., 27, 1449 (1992).
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Author:Kitagawa, Masayoshi; Zhou, Dexin; Qiu, Jianhui
Publication:Polymer Engineering and Science
Date:Nov 1, 1995
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