# Multiaxial ratcheting-fatigue interaction on acrylonitrile-butadiene-styrene terpolymer.

INTRODUCTION

Acrylonitrile-butadiene-styrene terpolymer (ABS) has been widely used to replace traditional materials in many applications due to its good mechanical properties, chemical resistance, and processing advantages [1-4]. Many studies have shown that ABS has high resistance to uniaxial or multiaxial impact [5-10],

As one of the most important characteristics of polymer is the decrease of ratcheting rate, the plastic shakedown capability, and the memory and recovery of the accumulated plastic strain under symmetrical stress cycles, the ratcheting behaviors, and fatigue predictions of polymer under uniaxial and multiaxial loadings have been studied by both experiments and simulations. Using experiments, Shen et al. [11] and Tao and Xia, [12, 13] studied the cyclic deformation of epoxy under cyclic/fatigue loading. The ratcheting behavior of epoxy and its effect on fatigue life was also discussed. Chen et al. [14-16] conducted a series of tests on vulcanized natural rubber and polytetrafluoroethene under cyclic compressive and multiaxial loads. The effects of mean stress, shear strain range, shear strain rate, and loading history on both two materials were discussed. It was found by Wang et al. [17] and Zhang and Chen [18] that temperature, as well as mean stress and stress amplitude, had a positive influence on ratcheting accumulation under uniaxial loading. Shariati et al. [19] presented an experimental study on the ratcheting behavior of polyoxymethylene (POM) under uniaxial cyclic loading. The effects of mean stress and stress rate on the ratcheting strain of POM were evaluated. Furthermore, mean stress functions were used as the equivalent damage parameter in the fatigue life predictions.

In addition, some fatigue life models were also proposed in recent years to give life predictions on polymers under uniaxial and multiaxial loading. Drozdov and Christiansen [20, 21] studied the cyclic viscoelastoplasticity of PA6 in tensile relaxation tests. A low-cycle fatigue model with adjustable parameters fitted by relaxation tests and cyclic tests was developed. Fatigue life models based on the Basquin law [22] was also used for life predictions of polymers. Bureau and Denault's studies on continuous glass fiber/polypropylene composites [23] show that the S-N curve and the Basquin law could give good fatigue life predictions on brittle polymers. Similar results were also obtained for epoxies [24] and PA6 nanocomposites [25], Unfortunately, most of the studies on ABS have been focused on the fatigue behaviors under uniaxial loading and its crack growing [26-29]. Studies on the multiaxial fatigue behavior of ABS are few, especially those on the influence of negative mean stress and the recovery of ratcheting strain under multistep loading conditions.

The current work aims to study the multiaxial ratcheting behavior of ABS. Particular attention is paid to experimentally evaluate the effect of loading history on the ratcheting behavior of ABS. We also propose a possible mechanism on how the loading history affects the ratcheting behavior of ABS in term of molecular structure. The results and discussions are useful for understanding the mechanism of ratcheting deformation of polymers and guiding the practical applications of ABS. Finally, a modified stress-based fatigue life model is presented to predict the fatigue life of ABS under pure torsional and multiaxial loadings.

MATERIALS AND METHODS

ASTM E2207-08 recommends a specimen thickness of 2 mm and diameter over 20 mm. Following this recommendation, the maximum tensile load and torsional torque to fracture the specimen are beyond the capability of our test machine. Referring other studies of the transport tube [30, 31], we used the industrial ABS fluent transport tubes as test specimens. The shape and dimensions of a typical test specimen are shown in Fig. 1. A pair of steel pillars was adhered to the inside of the specimen at each end to maintain the stability of the specimen during the test.

A series of experiments were performed using a CARE EUM-25k20 (Fig. 2) tension-torsion servo-electronic testing I machine, which has a capacity of [+ or -] 5 kN in the axial load and [+ or -] 20 N.m in the torque. A linear encoder and a rotary encoder are used to measure the axial displacement and torsional angle of the specimen, respectively. The fixtures are powered by hydraulic pressure to ensure that the force exerted on the specimen is constant and to prevent the specimen from getting loose.

The length L of the specimen between the hydraulic fixtures was measured before the test. The axial strain e was calculated from the relative displacement l of the specimen, which was detected by the linear encoder. The shear strain [gamma] was calculated from the torsional degree a measured by the rotary encoder. The axial strain and shear strain were obtained from Eqs. 1 and 2:

[epsilon] = l/L (1)

[gamma] = [alpha][pi]R/180L. (2)

The axial stress [sigma] and shear stress [tau] of the specimen were calculated by Eqs. 3 and 4:

[sigma] = R/[pi]([R.sup.2] - [r.sup.2]) (3)

[tau] = 2T/[pi](R + r)([R.sup.2] - [r.sup.2]) (4)

where R is the outer radius (mm) of specimen, r is the inner radius (mm) of specimen, and F and T are the load (N) and torque (N.m) detected by the axial-torsional load cell, respectively.

The ratcheting strain is defined as the average value of the maximum axial strain [[epsilon].sub.max] and minimum axial strain [[epsilon].sub.min] in each cycle and given by Eq. 5:

[[epsilon].sub.r] = [[epsilon].sub.max] + [[epsilon].sub.min]/2. (5)

The von Mises equivalent shear strain is obtained from Eq. 6:

[[epsilon].sub.eq] = [gamma]/[square root of 3]. (6)

RESULTS AND DISCUSSION

Static Mechanical Properties

Monotonic tension experiment on ABS under a constant stress rate of 0.1 MPa/s was conducted. The stress-strain curve is illustrated in Fig. 3a. The torsional test was conducted at a constant strain rate of [10.sup.-2]/s. The stress-strain curve is shown in Fig. 3b. All the monotonic experiments were performed until macrocracks appeared on the surface of the specimens. The static mechanical properties of ABS are summarized in Table 1.

Axial Ratcheting Behavior Under Pure Torsional Cyclic Loading

The pure torsional testing conditions and results of fatigue life under different shear strain amplitudes are listed in Table 2 (Specimen Nos. 1-6). The tests were performed under shear strain control at the constant shear strain rate of [10.sup.-2]/s. All the tests were performed on a multiaxial closed loop servo-electronic test machine. Load, displacement, torque, and degree of specimen during a test were monitored and controlled through a dynamic multiaxial controller. For pure torsional strain-controlled tests, the torsional degree control mode was used in the torsional direction while the load control mode was applied in the axial direction (zero mean stress and zero stress amplitude) to achieve the stress free status. Regardless of the kind of axial deformation of the specimen, as long as the axial stress was set to zero in advance, the axial stress will be automatically adjusted to zero because the closed loop load control mode was imposed in the axial direction. In these tests, positive axial strain accumulation, which is referred to as the ratcheting strain, was observed. Figure 4 shows a typical development of axial ratcheting strain under pure torsional strain cycling (Specimen No. 2).

Figure 5 (all the curve are shown from 1st to 500th cycles) shows that different shear strain amplitudes lead to different plastic strain and different accumulation rates of axial ratcheting strain. It can be observed that the axial ratcheting strain increases rapidly in the first 100 cycles and results in accumulation of 60-75% of the final strain. This stage is referred to as the transient stage [32], After this stage, the rate of ratcheting strain accumulation becomes slow, even stops. This second stage is referred to as the stabilized stage [32], As shown in Fig. 5, in the transient stage, which ends up at the 100th cycle, there is clearly a positive relationship between the equivalent shear strain amplitude and ratcheting strain accumulation. In the stabilized stage, however, the ratcheting strain accumulation rate gradually decreases to 1-5% of that in the transient stage at equivalent shear strain amplitudes smaller than 1.5%. As the equivalent shear strain amplitude increases to 1.75%, it becomes difficult to distinguish the transient stage and the stabilized stage.

Multiaxial Ratcheting Behavior

Multiaxial Ratcheting Under One-Step Loading. The results of one-step multiaxial tests are shown in Table 2 (Specimen Nos.7-11), in which the shear strain rate of [10.sup.-2]/s is the same as that used in pure torsional cyclic tests.

The effect of mean stress on ratcheting strain was obtained by comparing the results of Specimen Nos.4 and 7-11. All the tests were performed at same shear strain amplitude and shear strain rate but different axial mean stresses. The comparison of results is illustrated in Fig. 6, which shows the effect of the mean stress on multiaxial ratcheting strain accumulation. It can be concluded that the ratcheting strain accumulates more rapidly with the increase of tensile mean stress in the transient stage. As to the ratcheting strain rates in the stabilized stage, both values (estimated by dividing the ratcheting strain difference between the 200th and the 300th cycle by the time span of 100 cycles) under the mean stresses of 0 and 5 MPa are approximately 5 x [10.sup.-6] [s.sup.-1]. However, the ratcheting strain rate in the stabilized stage under the mean stress of 10 MPa is nearly 10 times larger than those under the mean stresses of 5 and 0 MPa. Furthermore, the compressive axial mean stress greatly reduces the ratcheting strain. The ratcheting strain decreases with the increase of compressive mean stress. After 300 cycles, the total ratcheting strain under the mean stress of -1 MPa is only 90% of that under the mean stress of 0 MPa, and the total ratcheting strain under the mean stress of -5 MPa is less than 40% of that under the mean stress of 0 MPa. The ratcheting strain stops accumulating at a compressive mean stress of -10 MPa. These results show that tensile mean stresses contribute to the ratcheting strain, whereas compressive mean stresses retard the ratcheting strain.

Multiaxial Ratcheting Under Multistep Loading. The loading history and ratcheting strain accumulation of loading Path 1 (Specimen No. 12) and Path II (Specimen No. 13) are shown in Fig. 7. Step 1 of Path I starts with an axial mean stress of 5 MPa and an equivalent shear strain amplitude of 1.0%. After 200 cycles, as the axial mean stress recovers to zero, the ratcheting strain decreases gradually. After another 200 cycles, the mean stress deceases to -5 MPa, and the ratcheting strain continues to decrease. Conversely, the mean stress in Step 2 of Path II directly decreases to -5MPa under the same loading history as Step 1 of Path I and leads to a greater negative ratcheting strain than those in Steps 2 and 3 of Path I. Then the mean stress recovers from -5 to 0 MPa after 200 cycles, which in turn increases the axial ratcheting strain in Step 3 of Path II.

The evolutions of ratcheting strain in Paths I and II are compared in Fig. 8. First, by comparing Step 2 of Path I and Step 3 of Path II, we can conclude that ratcheting strain relaxation, whether positive or negative, begins when the axial mean stress changes to 0 MPa. Furthermore, the negative relaxation of ratcheting strain is faster than the positive one under the same mean stress recovery (5 MPa).

Second, the combination of the negative axial ratcheting strain in Steps 2 and 3 of Path I are equal to the negative ratcheting strain in Step 2 of Path II. Actually, the loading history in Step 2 of Path II is also the loading histories in Steps 2 and 3 of Path I combined. As a result, the negative ratcheting strains in Steps 2 and 3 of Path I are independent of loading history.

Molecular Deformation Mechanism. The negative ratcheting strain is explained below by the displacement of ABS molecules.

An ABS molecule is represented by a Maxwell-type viscoelastic model [33], as illustrated in Fig. 9a. The model divides a single molecule into two parts: an elastic Gaussian network and a viscous Eyring process [34, 35], Thus, the total strain e consists of two parts: an elastic strain [[epsilon].sub.e] and a viscous strain [[epsilon].sub.v], as shown in Eq. 7.

d[epsilon] = d[[epsilon].sub.e] + d[[epsilon].sub.v] = d[sigma]/E + [sigma]/[eta] dt (7)

where E is the elasticity modulus and [eta] is the viscosity.

The stress/strain relaxation of polymer molecule [36] can be explained by the viscous flow in the Eyring process. As illustrated in Fig. 9b, an initial state is defined as the state before any loading history. When an ABS specimen is subjected to a large load, whether a tension or a compression, a strain consists of an elastic strain and a viscous strain results. Upon removal of the load, the elastic [[epsilon].sub.e] immediately recovers, but the viscous strain [[epsilon].sub.v] is unstable and relaxes slowly when the load recovers to its initial state. The value of strain relaxation in each cycle is defined as ([DELTA][sigma]/[eta])dt.

The negative ratcheting strain accumulation in Step 2 of Path I is illustrated in Fig. 10. At the beginning of the Step 1 of Path I, the length of molecule decreases as the mean stress decreases from 5 to 0 MPa. This strain decrease is as large as the elastic strain ([[epsilon].sub.e]).sub.tension] caused by mean stress in Step 1. It should be emphasized that after 200 cycles, the ratcheting strain accumulation caused by cyclic torsion is in the stabilized stage and can be, therefore, ignored here. The strain relaxation caused by viscous flow can be used here to explain the decrease of the ratcheting strain in Step 2 of Path I. As the axial stress recovers to 0 MPa, the viscous strain starts to relax to its initial state. The direction of ratcheting strain relaxation is the same as that of the axial stress recovery. After 200 cycles, the total ratcheting strain accumulates to [florin]([DELTA][sigma]/[eta]])dt and is negative. The negative ratcheting strain accumulation in Step 3 of Path I is also illustrated in Fig. 10. The compressibility of molecules is used here to explain the decrease of ratcheting strain. For each cycle, the viscous strain caused by compressive mean stress is defined as [([sigma]/[eta]).sub.compression]dt. By the end of cyclic loading, the total ratcheting strain accumulates to [florin] [([sigma]/[eta]).sub.compression]dt. Just like [florin]([DELTA][sigma]/[eta]/)dt, the ratcheting strain [florin] [([sigma]/[eta]).sub.compression]dt is also a negative number.

The negative ratcheting strain accumulation of ABS molecules in Step 2 of Path II is illustrated in Fig. 11. In terms of loading history, Step 2 of Path II is the combination of Steps 2 and 3 of Path I. The results given in Fig. 8 show that the ratcheting strain caused by strain relaxation and compressive mean stress are independent of loading history. Thus, under the same loading history as Step 1 of Path I, the mean stress in Path II directly reduces to -5 MPa and results two parts of negative ratcheting strain, ([DELTA][sigma]/[eta])dt and [([sigma]/[eta]).sub.compression]dt, as discussed above. We have shown that the negative ratcheting strain accumulation in this step is the superposition of the negative ratcheting strains caused by strain relaxation and compressive mean stress. The negative ratcheting strain in this step is as large as ([DELTA][sigma]/[eta]dt + dt + [([sigma]/[eta]).sub.compression]dt. After 200 cycles, the negative ratcheting strain accumulates to [florin]([DELTA][sigma]/[eta]])dt + [florin] [([sigma]/[eta]).sub.compression]dt

Fatigue Life Prediction

The results of fatigue life under different shear strain amplitudes are listed in Table 2. Fatigue life of a specimen is defined as the number of cycles taken for visible cracks to emerge. The stabilized stress-strain loops under different equivalent shear strain amplitudes are shown in Fig. 12. It can be obtained that the plastic strain amplitude of ABS is quite small, and the stabilized stress-strain loop of ABS is similar as the brittle material. As there are no clear-cut elastic and plastic stages during the loading process, it is difficult to distinguish the elastic and plastic parts of the damage. As a result, it is difficult to accurately fit the fatigue strength coefficients and the fatigue ductility coefficients of the Manson-Coffin law [37], Thus, critical plane models, such as the Smith-Watson-Topper law [38] and the fatigue damage coupled models [39], which are all based on the Manson-Coffin law, are not appropriate for predicting the fatigue life of ABS.

The Basquin law, a stress-life law, was used here to describe the relationship between shear stress amplitude and fatigue life of ABS. A low-cycle stress-life curve based on the Basquin law (Eq. 8) was fitted to the pure torsional results of the ABS samples (Table 2):

[DELTA][tau]/2 = [[tau]'.sub.f][([N.sub.f]).sup.b] (8)

where [[tau]'.sub.f] is the fatigue strength coefficient and b is the fatigue strength exponent, [[tau]'.sub.f] = 39.22 and b = -0.20. [N.sub.f] is the cycles number to failure. The results in Table 2 clearly show that the axial mean stress influences the fatigue life dramatically. The tensile mean stress reduces the fatigue life and the compressive mean stress increases the fatigue life. A multiaxial life model mostly based on the Basquin law but also considers the influence of mean stress is used here:

ESA = n x [[sigma].sub.mean] + [DELTA][tau]/2 = [[tau]'.sub.f][([N.sub.f]).sup.b] (9)

where ESA is the equivalent stress amplitude and n is the coefficient number, n = 0.2 was used to fit the results of the multiaxial tests shown in Fig. 13.

Figure 14 shows good agreement between the experimental data and the predictions taking into account the mean stress. In addition, Eq. 9 can also be used for pure torsional fatigue life predictions.

CONCLUSIONS

Ratcheting strain occurred and accumulated when ABS was subjected to pure symmetrical torsion or torsional loading conditions with axial stress. The axial strain accumulation could be divided into a transient stage and a stabilized stage. The axial ratcheting strain and its rate increased during the transient stage with increasing shear strain amplitude. When the equivalent shear strain amplitude increased up to 1.75%, it became difficult to distinguish the transient stage and the stabilized stage.

The axial mean stress affected the multiaxial ratcheting strain and the fatigue life. The tensile mean stress contributed to the ratcheting strain accumulation but reduced the fatigue life, while the compressive mean stress retarded the ratcheting strain accumulation and contributed to increase the fatigue life.

In the multistep tests, negative ratcheting strain was observed when tensile mean stress recovered from 5 to 0 MPa and can be explained as the strain relaxation caused by viscous strain How. When the specimens were subjected to a compressive mean stress, negative ratcheting strain was also found and can be explained as the compressive viscous strain. These two kinds of negative ratcheting strain are all independent of loading history.

Furthermore, a modified multiaxial fatigue life model taking into account the effect of mean stress was proposed to predict the fatigue life of ABS under complicated loading conditions. The predicted results agreed with the experimental results very well.

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Gang Chen, (1) Hao-qiong Liang, (1) Lei Wang, (1) Yun-hui Mei, (2,3) Xu Chen (1)

(1) School of Chemical Engineering and Technology, Tianjin University, Tianjin, China

(2) Tianjin Key Laboratory of Advanced Joining Technology, Tianjin University, Tianjin, China

(3) School of Materials Science and Engineering, Tianjin University, Tianjin, China

Correspondence to: Dr. Yun-hui Mei; e-mail: yunhui@tju.edu.cn

Contract grant sponsor: National Natural Science Foundation of China; contract grant numbers: 11172202; 51101112.

Contract grant sponsor: Program for New Century Excellent Talents in University (NCET-13-0400) of China.

DOI 10.1002/pen.23932

Published online in Wiley Online Library (wileyonlinelibrary.com).

Acrylonitrile-butadiene-styrene terpolymer (ABS) has been widely used to replace traditional materials in many applications due to its good mechanical properties, chemical resistance, and processing advantages [1-4]. Many studies have shown that ABS has high resistance to uniaxial or multiaxial impact [5-10],

As one of the most important characteristics of polymer is the decrease of ratcheting rate, the plastic shakedown capability, and the memory and recovery of the accumulated plastic strain under symmetrical stress cycles, the ratcheting behaviors, and fatigue predictions of polymer under uniaxial and multiaxial loadings have been studied by both experiments and simulations. Using experiments, Shen et al. [11] and Tao and Xia, [12, 13] studied the cyclic deformation of epoxy under cyclic/fatigue loading. The ratcheting behavior of epoxy and its effect on fatigue life was also discussed. Chen et al. [14-16] conducted a series of tests on vulcanized natural rubber and polytetrafluoroethene under cyclic compressive and multiaxial loads. The effects of mean stress, shear strain range, shear strain rate, and loading history on both two materials were discussed. It was found by Wang et al. [17] and Zhang and Chen [18] that temperature, as well as mean stress and stress amplitude, had a positive influence on ratcheting accumulation under uniaxial loading. Shariati et al. [19] presented an experimental study on the ratcheting behavior of polyoxymethylene (POM) under uniaxial cyclic loading. The effects of mean stress and stress rate on the ratcheting strain of POM were evaluated. Furthermore, mean stress functions were used as the equivalent damage parameter in the fatigue life predictions.

In addition, some fatigue life models were also proposed in recent years to give life predictions on polymers under uniaxial and multiaxial loading. Drozdov and Christiansen [20, 21] studied the cyclic viscoelastoplasticity of PA6 in tensile relaxation tests. A low-cycle fatigue model with adjustable parameters fitted by relaxation tests and cyclic tests was developed. Fatigue life models based on the Basquin law [22] was also used for life predictions of polymers. Bureau and Denault's studies on continuous glass fiber/polypropylene composites [23] show that the S-N curve and the Basquin law could give good fatigue life predictions on brittle polymers. Similar results were also obtained for epoxies [24] and PA6 nanocomposites [25], Unfortunately, most of the studies on ABS have been focused on the fatigue behaviors under uniaxial loading and its crack growing [26-29]. Studies on the multiaxial fatigue behavior of ABS are few, especially those on the influence of negative mean stress and the recovery of ratcheting strain under multistep loading conditions.

The current work aims to study the multiaxial ratcheting behavior of ABS. Particular attention is paid to experimentally evaluate the effect of loading history on the ratcheting behavior of ABS. We also propose a possible mechanism on how the loading history affects the ratcheting behavior of ABS in term of molecular structure. The results and discussions are useful for understanding the mechanism of ratcheting deformation of polymers and guiding the practical applications of ABS. Finally, a modified stress-based fatigue life model is presented to predict the fatigue life of ABS under pure torsional and multiaxial loadings.

MATERIALS AND METHODS

ASTM E2207-08 recommends a specimen thickness of 2 mm and diameter over 20 mm. Following this recommendation, the maximum tensile load and torsional torque to fracture the specimen are beyond the capability of our test machine. Referring other studies of the transport tube [30, 31], we used the industrial ABS fluent transport tubes as test specimens. The shape and dimensions of a typical test specimen are shown in Fig. 1. A pair of steel pillars was adhered to the inside of the specimen at each end to maintain the stability of the specimen during the test.

A series of experiments were performed using a CARE EUM-25k20 (Fig. 2) tension-torsion servo-electronic testing I machine, which has a capacity of [+ or -] 5 kN in the axial load and [+ or -] 20 N.m in the torque. A linear encoder and a rotary encoder are used to measure the axial displacement and torsional angle of the specimen, respectively. The fixtures are powered by hydraulic pressure to ensure that the force exerted on the specimen is constant and to prevent the specimen from getting loose.

The length L of the specimen between the hydraulic fixtures was measured before the test. The axial strain e was calculated from the relative displacement l of the specimen, which was detected by the linear encoder. The shear strain [gamma] was calculated from the torsional degree a measured by the rotary encoder. The axial strain and shear strain were obtained from Eqs. 1 and 2:

[epsilon] = l/L (1)

[gamma] = [alpha][pi]R/180L. (2)

The axial stress [sigma] and shear stress [tau] of the specimen were calculated by Eqs. 3 and 4:

[sigma] = R/[pi]([R.sup.2] - [r.sup.2]) (3)

[tau] = 2T/[pi](R + r)([R.sup.2] - [r.sup.2]) (4)

where R is the outer radius (mm) of specimen, r is the inner radius (mm) of specimen, and F and T are the load (N) and torque (N.m) detected by the axial-torsional load cell, respectively.

The ratcheting strain is defined as the average value of the maximum axial strain [[epsilon].sub.max] and minimum axial strain [[epsilon].sub.min] in each cycle and given by Eq. 5:

[[epsilon].sub.r] = [[epsilon].sub.max] + [[epsilon].sub.min]/2. (5)

The von Mises equivalent shear strain is obtained from Eq. 6:

[[epsilon].sub.eq] = [gamma]/[square root of 3]. (6)

RESULTS AND DISCUSSION

Static Mechanical Properties

Monotonic tension experiment on ABS under a constant stress rate of 0.1 MPa/s was conducted. The stress-strain curve is illustrated in Fig. 3a. The torsional test was conducted at a constant strain rate of [10.sup.-2]/s. The stress-strain curve is shown in Fig. 3b. All the monotonic experiments were performed until macrocracks appeared on the surface of the specimens. The static mechanical properties of ABS are summarized in Table 1.

Axial Ratcheting Behavior Under Pure Torsional Cyclic Loading

The pure torsional testing conditions and results of fatigue life under different shear strain amplitudes are listed in Table 2 (Specimen Nos. 1-6). The tests were performed under shear strain control at the constant shear strain rate of [10.sup.-2]/s. All the tests were performed on a multiaxial closed loop servo-electronic test machine. Load, displacement, torque, and degree of specimen during a test were monitored and controlled through a dynamic multiaxial controller. For pure torsional strain-controlled tests, the torsional degree control mode was used in the torsional direction while the load control mode was applied in the axial direction (zero mean stress and zero stress amplitude) to achieve the stress free status. Regardless of the kind of axial deformation of the specimen, as long as the axial stress was set to zero in advance, the axial stress will be automatically adjusted to zero because the closed loop load control mode was imposed in the axial direction. In these tests, positive axial strain accumulation, which is referred to as the ratcheting strain, was observed. Figure 4 shows a typical development of axial ratcheting strain under pure torsional strain cycling (Specimen No. 2).

Figure 5 (all the curve are shown from 1st to 500th cycles) shows that different shear strain amplitudes lead to different plastic strain and different accumulation rates of axial ratcheting strain. It can be observed that the axial ratcheting strain increases rapidly in the first 100 cycles and results in accumulation of 60-75% of the final strain. This stage is referred to as the transient stage [32], After this stage, the rate of ratcheting strain accumulation becomes slow, even stops. This second stage is referred to as the stabilized stage [32], As shown in Fig. 5, in the transient stage, which ends up at the 100th cycle, there is clearly a positive relationship between the equivalent shear strain amplitude and ratcheting strain accumulation. In the stabilized stage, however, the ratcheting strain accumulation rate gradually decreases to 1-5% of that in the transient stage at equivalent shear strain amplitudes smaller than 1.5%. As the equivalent shear strain amplitude increases to 1.75%, it becomes difficult to distinguish the transient stage and the stabilized stage.

Multiaxial Ratcheting Behavior

Multiaxial Ratcheting Under One-Step Loading. The results of one-step multiaxial tests are shown in Table 2 (Specimen Nos.7-11), in which the shear strain rate of [10.sup.-2]/s is the same as that used in pure torsional cyclic tests.

The effect of mean stress on ratcheting strain was obtained by comparing the results of Specimen Nos.4 and 7-11. All the tests were performed at same shear strain amplitude and shear strain rate but different axial mean stresses. The comparison of results is illustrated in Fig. 6, which shows the effect of the mean stress on multiaxial ratcheting strain accumulation. It can be concluded that the ratcheting strain accumulates more rapidly with the increase of tensile mean stress in the transient stage. As to the ratcheting strain rates in the stabilized stage, both values (estimated by dividing the ratcheting strain difference between the 200th and the 300th cycle by the time span of 100 cycles) under the mean stresses of 0 and 5 MPa are approximately 5 x [10.sup.-6] [s.sup.-1]. However, the ratcheting strain rate in the stabilized stage under the mean stress of 10 MPa is nearly 10 times larger than those under the mean stresses of 5 and 0 MPa. Furthermore, the compressive axial mean stress greatly reduces the ratcheting strain. The ratcheting strain decreases with the increase of compressive mean stress. After 300 cycles, the total ratcheting strain under the mean stress of -1 MPa is only 90% of that under the mean stress of 0 MPa, and the total ratcheting strain under the mean stress of -5 MPa is less than 40% of that under the mean stress of 0 MPa. The ratcheting strain stops accumulating at a compressive mean stress of -10 MPa. These results show that tensile mean stresses contribute to the ratcheting strain, whereas compressive mean stresses retard the ratcheting strain.

Multiaxial Ratcheting Under Multistep Loading. The loading history and ratcheting strain accumulation of loading Path 1 (Specimen No. 12) and Path II (Specimen No. 13) are shown in Fig. 7. Step 1 of Path I starts with an axial mean stress of 5 MPa and an equivalent shear strain amplitude of 1.0%. After 200 cycles, as the axial mean stress recovers to zero, the ratcheting strain decreases gradually. After another 200 cycles, the mean stress deceases to -5 MPa, and the ratcheting strain continues to decrease. Conversely, the mean stress in Step 2 of Path II directly decreases to -5MPa under the same loading history as Step 1 of Path I and leads to a greater negative ratcheting strain than those in Steps 2 and 3 of Path I. Then the mean stress recovers from -5 to 0 MPa after 200 cycles, which in turn increases the axial ratcheting strain in Step 3 of Path II.

The evolutions of ratcheting strain in Paths I and II are compared in Fig. 8. First, by comparing Step 2 of Path I and Step 3 of Path II, we can conclude that ratcheting strain relaxation, whether positive or negative, begins when the axial mean stress changes to 0 MPa. Furthermore, the negative relaxation of ratcheting strain is faster than the positive one under the same mean stress recovery (5 MPa).

Second, the combination of the negative axial ratcheting strain in Steps 2 and 3 of Path I are equal to the negative ratcheting strain in Step 2 of Path II. Actually, the loading history in Step 2 of Path II is also the loading histories in Steps 2 and 3 of Path I combined. As a result, the negative ratcheting strains in Steps 2 and 3 of Path I are independent of loading history.

Molecular Deformation Mechanism. The negative ratcheting strain is explained below by the displacement of ABS molecules.

An ABS molecule is represented by a Maxwell-type viscoelastic model [33], as illustrated in Fig. 9a. The model divides a single molecule into two parts: an elastic Gaussian network and a viscous Eyring process [34, 35], Thus, the total strain e consists of two parts: an elastic strain [[epsilon].sub.e] and a viscous strain [[epsilon].sub.v], as shown in Eq. 7.

d[epsilon] = d[[epsilon].sub.e] + d[[epsilon].sub.v] = d[sigma]/E + [sigma]/[eta] dt (7)

where E is the elasticity modulus and [eta] is the viscosity.

The stress/strain relaxation of polymer molecule [36] can be explained by the viscous flow in the Eyring process. As illustrated in Fig. 9b, an initial state is defined as the state before any loading history. When an ABS specimen is subjected to a large load, whether a tension or a compression, a strain consists of an elastic strain and a viscous strain results. Upon removal of the load, the elastic [[epsilon].sub.e] immediately recovers, but the viscous strain [[epsilon].sub.v] is unstable and relaxes slowly when the load recovers to its initial state. The value of strain relaxation in each cycle is defined as ([DELTA][sigma]/[eta])dt.

The negative ratcheting strain accumulation in Step 2 of Path I is illustrated in Fig. 10. At the beginning of the Step 1 of Path I, the length of molecule decreases as the mean stress decreases from 5 to 0 MPa. This strain decrease is as large as the elastic strain ([[epsilon].sub.e]).sub.tension] caused by mean stress in Step 1. It should be emphasized that after 200 cycles, the ratcheting strain accumulation caused by cyclic torsion is in the stabilized stage and can be, therefore, ignored here. The strain relaxation caused by viscous flow can be used here to explain the decrease of the ratcheting strain in Step 2 of Path I. As the axial stress recovers to 0 MPa, the viscous strain starts to relax to its initial state. The direction of ratcheting strain relaxation is the same as that of the axial stress recovery. After 200 cycles, the total ratcheting strain accumulates to [florin]([DELTA][sigma]/[eta]])dt and is negative. The negative ratcheting strain accumulation in Step 3 of Path I is also illustrated in Fig. 10. The compressibility of molecules is used here to explain the decrease of ratcheting strain. For each cycle, the viscous strain caused by compressive mean stress is defined as [([sigma]/[eta]).sub.compression]dt. By the end of cyclic loading, the total ratcheting strain accumulates to [florin] [([sigma]/[eta]).sub.compression]dt. Just like [florin]([DELTA][sigma]/[eta]/)dt, the ratcheting strain [florin] [([sigma]/[eta]).sub.compression]dt is also a negative number.

The negative ratcheting strain accumulation of ABS molecules in Step 2 of Path II is illustrated in Fig. 11. In terms of loading history, Step 2 of Path II is the combination of Steps 2 and 3 of Path I. The results given in Fig. 8 show that the ratcheting strain caused by strain relaxation and compressive mean stress are independent of loading history. Thus, under the same loading history as Step 1 of Path I, the mean stress in Path II directly reduces to -5 MPa and results two parts of negative ratcheting strain, ([DELTA][sigma]/[eta])dt and [([sigma]/[eta]).sub.compression]dt, as discussed above. We have shown that the negative ratcheting strain accumulation in this step is the superposition of the negative ratcheting strains caused by strain relaxation and compressive mean stress. The negative ratcheting strain in this step is as large as ([DELTA][sigma]/[eta]dt + dt + [([sigma]/[eta]).sub.compression]dt. After 200 cycles, the negative ratcheting strain accumulates to [florin]([DELTA][sigma]/[eta]])dt + [florin] [([sigma]/[eta]).sub.compression]dt

Fatigue Life Prediction

The results of fatigue life under different shear strain amplitudes are listed in Table 2. Fatigue life of a specimen is defined as the number of cycles taken for visible cracks to emerge. The stabilized stress-strain loops under different equivalent shear strain amplitudes are shown in Fig. 12. It can be obtained that the plastic strain amplitude of ABS is quite small, and the stabilized stress-strain loop of ABS is similar as the brittle material. As there are no clear-cut elastic and plastic stages during the loading process, it is difficult to distinguish the elastic and plastic parts of the damage. As a result, it is difficult to accurately fit the fatigue strength coefficients and the fatigue ductility coefficients of the Manson-Coffin law [37], Thus, critical plane models, such as the Smith-Watson-Topper law [38] and the fatigue damage coupled models [39], which are all based on the Manson-Coffin law, are not appropriate for predicting the fatigue life of ABS.

The Basquin law, a stress-life law, was used here to describe the relationship between shear stress amplitude and fatigue life of ABS. A low-cycle stress-life curve based on the Basquin law (Eq. 8) was fitted to the pure torsional results of the ABS samples (Table 2):

[DELTA][tau]/2 = [[tau]'.sub.f][([N.sub.f]).sup.b] (8)

where [[tau]'.sub.f] is the fatigue strength coefficient and b is the fatigue strength exponent, [[tau]'.sub.f] = 39.22 and b = -0.20. [N.sub.f] is the cycles number to failure. The results in Table 2 clearly show that the axial mean stress influences the fatigue life dramatically. The tensile mean stress reduces the fatigue life and the compressive mean stress increases the fatigue life. A multiaxial life model mostly based on the Basquin law but also considers the influence of mean stress is used here:

ESA = n x [[sigma].sub.mean] + [DELTA][tau]/2 = [[tau]'.sub.f][([N.sub.f]).sup.b] (9)

where ESA is the equivalent stress amplitude and n is the coefficient number, n = 0.2 was used to fit the results of the multiaxial tests shown in Fig. 13.

Figure 14 shows good agreement between the experimental data and the predictions taking into account the mean stress. In addition, Eq. 9 can also be used for pure torsional fatigue life predictions.

CONCLUSIONS

Ratcheting strain occurred and accumulated when ABS was subjected to pure symmetrical torsion or torsional loading conditions with axial stress. The axial strain accumulation could be divided into a transient stage and a stabilized stage. The axial ratcheting strain and its rate increased during the transient stage with increasing shear strain amplitude. When the equivalent shear strain amplitude increased up to 1.75%, it became difficult to distinguish the transient stage and the stabilized stage.

The axial mean stress affected the multiaxial ratcheting strain and the fatigue life. The tensile mean stress contributed to the ratcheting strain accumulation but reduced the fatigue life, while the compressive mean stress retarded the ratcheting strain accumulation and contributed to increase the fatigue life.

In the multistep tests, negative ratcheting strain was observed when tensile mean stress recovered from 5 to 0 MPa and can be explained as the strain relaxation caused by viscous strain How. When the specimens were subjected to a compressive mean stress, negative ratcheting strain was also found and can be explained as the compressive viscous strain. These two kinds of negative ratcheting strain are all independent of loading history.

Furthermore, a modified multiaxial fatigue life model taking into account the effect of mean stress was proposed to predict the fatigue life of ABS under complicated loading conditions. The predicted results agreed with the experimental results very well.

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Gang Chen, (1) Hao-qiong Liang, (1) Lei Wang, (1) Yun-hui Mei, (2,3) Xu Chen (1)

(1) School of Chemical Engineering and Technology, Tianjin University, Tianjin, China

(2) Tianjin Key Laboratory of Advanced Joining Technology, Tianjin University, Tianjin, China

(3) School of Materials Science and Engineering, Tianjin University, Tianjin, China

Correspondence to: Dr. Yun-hui Mei; e-mail: yunhui@tju.edu.cn

Contract grant sponsor: National Natural Science Foundation of China; contract grant numbers: 11172202; 51101112.

Contract grant sponsor: Program for New Century Excellent Talents in University (NCET-13-0400) of China.

DOI 10.1002/pen.23932

Published online in Wiley Online Library (wileyonlinelibrary.com).

TABLE 1. Static mechanical properties of ABS. Elasticity modulus (E) 850 MPa Shear modulus (G) 490 MPa Yield stress in tension (0.2% offset) ([sigma] y) 15.1 MPa Ultimate tensile strength ([sigma] u) 32.1 MPa Elongation ([delta] 5) 8.1% TABLE 2. Symmetrical strain-controlled cyclic torsional test results. [[sigma].sub.mean] [DELTA][[gamma].sub.eq]/2 Spec. No (MPa) (%) 1 0 1.75 2 0 1.5 3 0 1.25 4 0 1 5 0 0.8 6 0 0.7 7 10 1 8 5 1 9 -1 1 10 -5 1 11 -10 1 12(Path I) 5 1 0 -5 13(Path II) 5 1 -5 0 [[sigma].sub.mean] Shear strain Fatigue life Spec. No (MPa) rate ([s.suup.-1]) (Cycles) 1 0 [10.sup.-2] 178 2 0 [10.sup.-2] 378 3 0 [10.sup.-2] 670 4 0 [10.sup.-2] 1381 5 0 [10.sup.-2] 4545 6 0 [10.sup.-2] 15621 7 10 [10.sup.-2] 392 8 5 [10.sup.-2] 768 9 -1 [10.sup.-2] 1768 10 -5 [10.sup.-2] 3656 11 -10 [10.sup.-2] 400 (a) 12(Path I) 5 [10.sup.-2] 200 (a) 0 200 (a) -5 200 (a) 13(Path II) 5 [10.sup.-2] 200 (a) -5 200 (a) 0 200 (a) (a) indicates no failure in test. The loading histories of Spec Nos. 12 and 13 are defined as Paths I and II, respectively.

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Author: | Chen, Gang; Liang, Hao-qiong; Wang, Lei; Mei, Yun-hui; Chen, Xu |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Date: | Mar 1, 2015 |

Words: | 4238 |

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