The Residual Stress Relaxation Behavior of Weldments During Cyclic Loading

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Materials Science and Engineering Publications

Materials Science and Engineering

7-2013

The Residual Stress Relaxation Behavior of

Weldments During Cyclic Loading

Zhongyuan Qian

Iowa State University

L. Scott Chumbley

Iowa State University, chumbley@iastate.edu

Tugce Karakulak

tugcek@iastate.edu

Eric Johnson

John Deere

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The Residual Stress Relaxation Behavior of Weldments During

Cyclic Loading

ZHONGYUAN QIAN, SCOTT CHUMBLEY, TUGCE KARAKULAK, and ERIC JOHNSON

Accurate measurement of residual stress is necessary to obtain reliable predictions of fatigue lifetime and enable estimation of time-to-facture for any given stress level. In this article, relaxation of welding residual stresses as a function of cyclic loading was documented on three common steels: AISI 1008, ASTM A572, and AISI 4142. Welded specimens were subjected to cyclic bending (R = 0.1) at different applied stresses, and the residual stress relaxation existing near the welds was measured as a function of cycles. The steels exhibited very different stress relaxation behaviors during cyclic loadings, which can be related to the differences in the microstructures of the specimens. A phenomenological model, which treats dislocation motion during cyclic loading as being analogous to creep of dislocations, is proposed for estimation of the residual stress relaxation.

DOI: 10.1007/s11661-013-1688-9

Ó The Minerals, Metals & Materials Society and ASM International 2013

I. INTRODUCTION

R

ESIDUAL stresses introduced during the welding process can have a signiﬁcant inﬂuence on the fatigue strength of welded components.[1–7] There are many ways to reduce potentially harmful residual stress to prolong service life, such as annealing,[8,9] pre-stretch-ing,[6]and shakedown.[10–12]For the purpose of accurate assessment of fatigue lifetime, not only the magnitude of the residual stresses, but also their stability factors and evolution during service are of great importance.[13]

Numerous researchers have reported significant relax-ation during cyclic loading,[7,14–24]with redistribution of residual stresses occurring after the first loading cycle followed by minimal further relaxation during subse-quent cycles.[8,25–27] By applying the principles of con-tinuum mechanics and fundamental physics, the evolution and redistribution of residual stresses such as those caused by welding can be explained.[28] How-ever, an understanding of the complicated interactions that result between the applied stresses and those that develop when a part is welded requires knowledge of a full 3D residual stress profile and detailed material properties, which is rarely possible in practice. As an alternative, some empirical relaxation models[25,27,29–31] based on experimental data obtained from studies of a number of different combinations of material and loading have been proposed. For example, Valluri

et al.[32] suggested that residual stress relaxation is a phenomenon similar to creep. Several investiga-tions[8,27,33] have shown that the amount of relaxation is a function of applied cyclic stress. However, one common deﬁciency of the existing models is that they are unable to separate variables related to the material from those related to the applied stresses. Perhaps even more importantly, they are incapable of accounting for the high amount of relaxation that occurs during the ﬁrst cycle. These two aspects were emphasized in the current study. In general, the proposed mechanisms for residual stress relaxation due to cyclic loading can be divided into three regimes according to the applied cyclic stress amplitudes: (1) below the endurance limit, (2) above the macroscopic yield strength, and (3) in between these two extremes.[13]

In the current study, cyclic amplitudes were applied in all three regimes noted above on different weld speci-mens, including low carbon steel AISI 1008, low alloy steel ASTM A572 and AISI 4142. These steels were selected for study because of their wide use in engineer-ing applications such as structural components, heavy construction equipment, and machinery parts and components. The steels selected range from low-to-medium strength. The corresponding relaxation behav-iors of the residual stresses present in welded samples subjected to cyclic loading were measured by X-ray diffraction method,[34–38] and the results obtained are presented later. These results are related to the observed microstructures and are discussed in light of possible dislocation movement in the materials. A phenomeno-logical model analogous to the mechanism of disloca-tion creep, which correlates the amount of relaxadisloca-tion with the applied stress, number of cycles, magnitude of welding stress, and several material variables, is pro-posed for the first time and compared with the experimental results.

ZHONGYUAN QIAN, formerly Ph.D. Candidate with the Iowa State University, Ames, IA 50011, is now Materials Consultant with the Invetech Inc., Houston, TX. Contact e-mail: zqian@invetech.com SCOTT CHUMBLEY, Professor, and TUGCE KARAKULAK, Ph.D. Candidate, are with the Iowa State University. ERIC JOHNSON, Staﬀ Engineer, is with the John Deere Company, Moline, IL.

Manuscript submitted May 2, 2012. Article published online March 15, 2013

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II. EXPERIMENTS A. Materials and Specimens

Materials chosen for the current study were low carbon steel AISI 1008 and low alloy steels ASTM A572 and AISI 4142. These materials were selected because they are widely used in agriculture and construction machinery where the materials are often subjected to repeated loadings. The yield strength and ultimate tensile strength of the materials were measured and are shown in TableI.

The original stress state present in the as-received thin plate was ﬁrst determined by averaging several mea-surements, and those values are also shown in TableI. Note that the rolling of the sheet plate used in the current study results in an original compressive state in each of the materials and the stresses in longitudinal and transverse direction are almost equal.

Long plates (600 mm 9 100 mm, with slightly diﬀer-ing thicknesses) of all three materials were metal-inert-gas (MIG) welded using welding robots at (1) 21.7 V, 137.5 A, and welding speed 0.8 m/minute for AISI 1008 steel; and (2) 24 V, 248.5 A, and welding speed 0.7 m/minute for ASTMA572 and AISI 4142 materials, to make single pass bead on plate weld. The welded plates were sectioned into 50 mm 9 100 mm specimens for residual stress measurements. The dimen-sions of the specimens are shown in Table I and Figure 1, and a thin plate assumption was applied in the current study.

B. Experimental Procedures

A Bruker D-8 X-ray diﬀractometer, operated at 30 kV and 50 mA with a Cr tube, and with 0.5-mm circular collimator and 2D area detector, was used to measure the residual stresses. The gauge volume is approximately 1 9 104mm3. The stresses were mea-sured by the W method and determined using the a-Fe (211) peak shift according to SAE E915 standard.[39] The preliminary test shows the excellent reproducibility of less than ±9 MPa error. The measurements were taken as close to the weld root as possible (2 mm from the weld toe), in both transverse (perpendicular to the weld) and longitudinal (parallel to the weld) directions. On each specimen, three points (5 mm apart) were measured and averaged to reduce statistical errors, as shown in Figure1.

First, the welding residual stresses were measured after welding process. Then, cyclic four point bending was applied to the specimens along the transverse direction using a MTS servo-hydraulic fatigue-testing

machine according to the ASTM E855 standards. The measurement points were under tensile loadings at a loading ratio R = 0.1. The applied ﬂexural stress was calculated by the following Eq. [1]:

rs¼

3Fa

wt2 ½1

where F is the loading force, a is distance from the support to the load applicator, w is specimen width, and tis the specimen thickness.

After the first cycle, each specimen was removed from the fixture and taken for X-ray measurement. After measurement, the specimen was put back into the bending fixture for further loading. After ten cycles, each specimen was again taken for X-ray measurement at the exact same locations. Depending on the specific sample studied, this procedure was repeated at 100, 1000, and/or 10000 loading cycles, as shown in TableII. Several levels of loading, also shown in Table II, were applied to each material based on percentages of their yield strength. This insured that the three regimes of interest, namely (1) below the endurance limit, (2) above the yield strength, and (3) in between, were all covered. Each loading was repeated on at least three specimens, and the results were averaged to ensure statistical significance. After these tests, a select number of specimens were subjected to further increased loadings, the details of which are also included in TableII. The stress concentration factor (Kt) was calculated on the

welding geometries using ﬁnite-element models. The value of Kt at the locations of the residual stress

measurement is around 1.0 to 1.2.

One great advantage of the non-destructive nature of X-ray diﬀraction is that residual stress measurements

Table I. Properties of Test Materials Yield Strength (MPa) Tensile Strength (MPa) Original Residual Stress (MPa) Length (mm) Width (mm) Thickness (mm) Number of Specimens AISI 1008 253 341 73 100 50 3.88 22

ASTM A572 (thin) 345 449 51 100 50 4.0 6

ASTM A572 (thick) 345 449 98 100 50 6.0 6

AISI 4142 768 947 100 100 50 4.76 7

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could be repeated at the same locations on the samples after loading. As a result, any changes caused by diﬀerent loadings could be readily measured.

C. Experimental Results

1. Low carbon steel AISI 1008

A typical redistribution of residual stress on AISI 1008 steel is shown in Figure 2. The specimen was ﬁrst

subjected to a loading of 63 MPa, which is essentially equal to a quarter of the measured yield strength (0.25ry) of the material, up to 10000 cycles. The open

symbols show the results of three measured points (left, middle, and right as indicated in Figure1) on the specimen, which are 5 mm apart. The average value over these three points, indicated by the diamond symbols, was used to represent the stress state in this specimen. After the initial loading and cycles, the stress was raised to 167 MPa (0.66ry), and further cyclic loading was

applied up to 1000 cycles. These results are included in Figure2 at the right of the vertical dashed line. Note that error bars are only shown on one set of data to keep the plot from becoming too cluttered.

Throughout the current study, the same procedures as above were repeated for each loading on at least three specimens, and the measured results were averaged to ensure statistical signiﬁcance, producing a series of plots for each material and stress level tested. Several studies of weldments[40,41] reveal that in many cases fatigue cracks tends to open in a direction along the weld toe, which is caused by the superimposition of the applied load and transverse residual stress. Therefore, the stress in the transverse direction of a weld is typically of more concern in practice.

The results of relaxation under different loading conditions for the AISI 1008 are summarized in Figure3. In the transverse direction (Figure3(a)), significant relaxation was only observed after the first cycle. No change of the residual stress was seen with additional cycles, even when the loadings were increased after 10000 cycles, and linear fits of the data after the initial cycle showed only very slight slopes. The longitudinal stresses (Figure3(b)) showed a slightly different behav-ior in that no initial drop was seen. Linear fitting again revealed similar gentle slopes for all four applied loads, which indicated that only minor changes occurred throughout the loading cycles, even when an increase in loading stresses was applied. Thus, the longitudinal residual stress appears to remain at a constant magni-tude regardless of the amplimagni-tude or cycles.

2. Low alloy steel ASTM A572

Similar experiments were performed on both 4- and 6-mm thicknesses of ASTM A572 steel specimens under the applied stresses of 100 and 400 MPa, and the results are shown in Figure4. In the transverse direction, a large amount of relaxation was observed for both thicknesses after the ﬁrst cycle for the 400 MPa loadings,

Table II. Cyclic Loading Applied to the Specimens

Material Load (MPa) Cycles (N) Further Load (MPa) Further Cycles (N)

AISI 1008 63 0,1,10,100,1000,10000 167 1,10,100,1000 167 0,1,10,100,1000,10000 250 1,10,100,1000 250 0,1,10,100,1000,10000 300 1,10,100,1000 300 0,1,10,100,1000,10000 ASTM A572 100 0,1,10,100 400 1,10,100 400 0,1,10,100 AISI 4142 150 0,1,10,100 600 1,10,100 600 0,1,10,100 0 (weld) 1 10 100 1000 10000 further 1 10 100 1000 0 40 80 120 Number of Cycles (N)

Transverse Residual Stress (MPa)

Transverse Residual Stress Redistribution left middle right average

0 (weld) 1 10 100 1000 10000 further 1 10 (b) (a) 100 1000 0 40 80 120 160 Number of Cycles (N)

Longitudinal Residual Stress (MPa)

Longitudinal Residual Stress Redistribution

left middle right average

Fig. 2—Typical residual stress redistribution of one AISI 1008 speci-men 2E, under 63 MPa cyclic loading. The specispeci-men was subjected to a further loading of 167 MPa. (a) Transverse stress. (b) Longitu-dinal stress.

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followed by the stress remaining relatively constant. This differed greatly from the 100 MPa loadings, which showed no significant changes up to 100 cycles. At this point, the loading level was increased to 400 MPa, and a considerable amount of relaxation was seen after one cycle at this higher load. The amount of relaxation was on the same order of magnitude as occurred in the first cycle for specimens that were directly loaded to 400 MPa. Again, only a minor decrease was seen in the longitudinal stresses when the specimens were subjected to a stress amplitude that exceeded the yield strength.

Measurements showed that no relaxation occurred when the applied stress ra< ry, and a large relaxation

was observed when ra> ry. This relaxation behavior

has been widely observed also in the literature.[13,25] 3. Low alloy steel AISI 4142

The original AISI 4142 plate had a compressive residual stress of100 MPa. Figure5shows the measured

transverse and longitudinal residual stresses as a func-tion of cyclic loading after welding. Measurements show an average stress of 600 MPa in the longitudinal direction, which approximates the yield strength of the material.

In Figure5, there is no signiﬁcant stress relaxation in either transverse direction or longitudinal direction under the 150 MPa load. A minor stress reduction within the range of measurement error did occur after the initial cycle for the 600 MPa load. An experiment was attempted where after the 100th cycle, the loading was increased from 600 to 700 MPa for one specimen. Despite the specimen being below the bulk yield strength of 768 MPa, it failed right at the weld toe during the ﬁrst cycle, indicating that the local stress (where stress cannot be measured by x-ray because of geometrical constraints associated with the sample stage) is most likely even higher than the obtained measurements. 0 (weld) 1 10 100 1000 10000 further 1 10 (a) (b) 100 1000 0 20 40 60 80 100 Number of Cycles (N) T rans v er s e R e s idual S tr e s s ( M P a )

Transverse Residual Stress on AISI1008

63MPa 167MPa 250MPa 300MPa 0 (weld) 1 10 100 1000 10000 further 1 10 100 1000 0 20 40 60 80 100 120 140 Number of Cycles (N) Longi tudi n al R e s idual S tr e s s ( M P a )

Longitudinal Residual Stress on AISI1008

63MPa 167MPa 250MPa 300MPa

Fig. 3—The residual stress redistribution under diﬀerent loading lev-els on AISI 1008 steel. (a) Transverse stress. (b) Longitudinal stress. The experimental errors (±9 MPa) are not shown here for the sake of clarity. 0 (weld) 1 10 (a) (b) 100 further 1 10 100 -10 0 10 20 30 40 50 60 Number of Cycles (N)

Transverse Residual Stress in A572

100MPa (4mm) 400MPa (4mm) 100MPa (6mm) 400MPa (6mm) 0 (weld)0 1 10 100 20 40 60 80 100 Number of Cycles (N)

Longitudinal Residual Stress in A572

400MPa (4mm) 400MPa (6mm)

Fig. 4—The residual stress redistribution on 4- and 6-mm-thick A572 steel. (a) Transverse stress. (b) Longitudinal stress. For the sake of clarity, the experimental errors (±7 MPa) are not shown here.

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III. DISCUSSION

A. Determination of Residual Stress Relaxation

The experimental results for all three steels studied show that no obvious stress relaxation occurs in the longitudinal direction for a load applied perpendicular to the weld, while residual stress relaxation did occur in the direction transverse to the weld. These results are complementary to the earlier studies of Takanashi[25] and Holzapfel,[8] who applied loads in the longitudinal direction (i.e., parallel to the weld) and saw relaxation of the longitudinal stresses but not the transverse stresses. Measurable relaxation only occurred in the ﬁrst few cycles, and the largest amount of relaxation was always observed after the ﬁrst cycle. Minor changes of residual stress occurred during subsequent cycles conducted using the same loading, although their amounts were much smaller. This is consistent with results reported by several studies.[26,27]

Although the transverse residual stress of all three materials showed relaxation, the manner by which relaxation was produced was very different, with each showing its own distinctive relaxation behavior. For example, for plain carbon steel AISI 1008, residual stress relaxation occurred at every applied load, even for the very low loading of only a quarter of yield strength. This was not seen for the other steels studied. In order to compare results between steels, one must first consider the factors that contribute to the measured stress results. The hot-rolled AISI 1008 steel plate showed an average compressive residual stress of 73 MPa. This is the original plate stress before welding, while the welding stress is the measured stress after welding but before cyclic loading begins, ranging from 40 to 100 MPa. Even though welding is a continuous process, random fluctu-ations such as power input, chemical composition and/or texture of the sample, local restraints. etc., will cause variations of the welding residual stress distribution throughout the specimens. This was why a spread of measurements was made, and an average value was used in the plots displayed in Figures3,4, and5.

The percentage of relaxation, S, was calculated using the following expression and is illustrated in Figure6.

percentage of relaxation of Nth cyclesðSÞ ¼welding stress residual stress of Nth cycles

welding stress plate stress 100% ½2 The plate stress is the stress measured as existing in the as-received plate before welding. This value is used as the lower limit because it is the dominant stress throughout the specimen, and previous studies have shown that the residual stress appears to relax toward this value instead of zero.[42]

While using the percentage reduction facilitates com-parison of the obtained data, it does present a drawback in that the relaxation as deﬁned by Eq. [2] results in increased error when the welding stress is close to the original plate stress. For example, in the case of AISI

0 (weld) 1 10 100 further 1 10 (a) (b) 100 -80 -60 -40 -20 0 20

Number of Cycles (N) Transverse Residual Stress in AISI4142

150MPa 600MPa 0 (weld) 1 10 100 further 1 10 100 0 100 200 300 400 500 600 700 Number of Cycles (N)

Longitudinal Residual Stress in AISI4142

150MPa

Fig. 5—The residual stress redistribution in AISI 4142 steel. (a) Transverse stress. (b) Longitudinal stress.

Fig. 6—Illustration of the amount of residual stress relaxation. Plate stress is the original stress state of plate before welding. Welding stress is the stress caused by welding before cyclic loading was applied.

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4142 steel at 600 MPa loading (Figure 6(a)), the error in the calculation can be as large as ±15 pct. This high amount of error precludes meaningful interpretation of the relaxation behavior. For this reason, the plot for AISI 4142 will not be included in the following discussion.

1. AISI 1008

Turning now to a consideration of the data, the calculated amounts of percent relaxation for AISI 1008 steel are plotted in Figure7. Relaxation is seen at all loading levels, although the highest loading (300 MPa, 1.2ry) resulted in the largest amount. Relaxation of

residual stress at the lowest loading (63 MPa, 0.25ry) is

surprising and has rarely been reported.[43]

Increasing loading after 10,000 cycles did not alter the residual stress, which indicates that the stress has been (and can only be) relaxed during the first cycle. In other words, the amount of relaxation is solely dependent on the applied stress level of the first cycle. Thus, a high stress applied at the early stages of service or a severe preloading will be most effective in relieving detrimental transverse residual stress in AISI 1008 weldments.

2. ASTM 572

The ASTM A572 steel showed a more commonly reported behavior. A high loading that exceeded the yield point caused a relaxation in the residual stress, while low applied loads did not result in any notable changes. However, when specimens that initially had a low load applied were subjected to further loading, involvement of a higher stress relaxation was seen. The amount of relaxation was calculated in the same way as for the AISI 1008 steel and is shown in Figure 8. The residual stress values of the original plates are 51 and 98 MPa for 4- and 6-mm-thickness plates, respec-tively. It is worth noticing that the initial high loading (400 MPa) and low–high loading (100 MPa then 400 MPa) produce a similar total amount of relaxation of residual stress. Both loading proﬁles resulted in 20 pct relaxation. These results indicate that only a high loading beyond the yield strength of the material has any eﬀect on residual stress relaxation in this material.

Similar experimental results were obtained for spec-imens of both 4- and 6-mm thickness. Relaxation seems be independent on the thickness of the material but on the relationship of the applied stress to the yield strength. The high applied stresses cause plastic ﬂow and change local constraints of the residual stress, resulting in a redistribution/relaxation. This phenome-non can be explained by the theory of continuum mechanics.[28]

3. AISI 4142

AISI 4142 steel exhibited a behavior opposite in nature to that of AISI 1008 steel. No significant reduction of residual stress could be confirmed at any level of loading since the slight drop seen after one cycle for the 600 MPa initial load was within experimental error. Increasing loading to 600 MPa after initial cycles at 150 MPa further confirms this observation—the

small drop that may be seen in Figure5 being again well within experimental error. If one grants that slight drops are seen after the initial cycle upon loading to 600 MPa as shown in Figure5(a), it is interesting to note that in the transverse measurements where com-pressive stresses are found, the material does not relax to a value tending toward zero but become more compres-sive. Considering the compressive stress of the original plate, the residual stress appears to be relaxing toward the original state of 100 MPa (compression). The longitudinal stress showed almost no change for both directions, stay at a high value throughout the test. Again, the small change in value results in large errors when considering percentage relaxation so these values are not plotted.

B. Microstructures and Dislocations

Several studies[12,44] present evidence exists linking residual stress relaxation behavior to the microstruc-tures/dislocation density of the material. For example,

0 (weld) 1 10 100 1000 10000 further 1 10 100 1000 0 4 8 12 16 20 Number of Cycles (N) Amount of Relaxation (%)

Amount of Stress Relaxation in AISI1008

63MPa 167MPa 250MPa 300MPa

Fig. 7—The amount of relaxation in transverse stress on AISI 1008 steel. 0 (weld) 1 10 100 further 1 10 100 -10 0 10 20 Number of Cycles (N) A m o u n t o f R e la x a ti o n ( % )

Amount of Stress Relaxation in ASTM A572

100MPa (4mm) 400MPa (4mm) 100MPa(6mm) 400MPa(6mm)

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Walker et al.[12] found a correlation between evolution of residual stress and dislocation density on cold-rolled mild steel EN3b. It is interesting and unusual that the residual stress relaxation in AISI 1008 occurred at every applied load, even very low ones, while no significant changes happened in the AISI 4142 steel. Moreover, a further loading did not result in any changes for either material. It is believed that the observed differences in stress relaxation behaviors are related to the micro-structures of the materials that result because of welding. Welding is an intense local heat treatment, producing severe thermal gradients resulting in non-uniform plastic deformation. As a result, thermal misfits and local shrinkage/expansion can be the source of large amounts of dislocations as the material cools. The manner by which dislocation motion may be affected because of the varying microstructures is discussed below.

The microstructures of the materials at the locations where the residual stress measurements were taken are shown in Figure9. AISI 1008 has the microstructure of almost pure ferrite, with a large grain size of 10 to 15 lm. The ASTM A572 is composed primarily of ferrite with small amounts of pearlite, and the grain size

is less than 5 lm. AISI 4142 steel was primarily ﬁne pearlite and ferrite. A number of ﬁne precipitates smaller than 1 lm are also present.

One hypothesis for the results seen in AISI 1008 is that the large grains of essentially single-phase ferrite have little resistance to dislocation slip and/or climb. Dislocation movement at highly plastically deformed locations caused by welding may be expected to become activate, even under a very low stress, for a microstruc-ture such as this. This changes the local plastic con-straints and results in residual stress redistribution. Since the dislocations can move freely, the stress redistribution is completed within the ﬁrst few cycles and no further relaxation occurs even when increasing the applied load. Other studies[13] have suggested some forms of stress concentration such as microscopic yielding at the grain boundaries occurs, resulting in microplastic deformation. While ASTM A572 has a similar microstructure with AISI 1008 the presence of pearlite and a smaller grain size act as barriers to stop dislocation movement. Consequently, ASTM A572 shows no stress relaxation until these barriers are overcome by applying a loading that exceeds the material’s yield point.

Fig. 9—Microstructures taken at the locations of residual stress measurements. Etched by 2 pct nital. (a) AISI 1008. (b) ASTM A572. (c) AISI 4142.

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The 4142 steel has a very different microstructure from the others, consisting of fine pearlite particles and small precipitates, both of which can be expected to pin and tangle dislocations during cyclic loading, making the movement of dislocations extremely difficult. As a result, residual stress redistribution/relaxation is difficult to achieve even when a load is applied near or at the yield stress. A higher applied loads might be able to activate dislocation movements and result in stress relaxation; however, when this was attempted the material failed. Two of the specimens were also exam-ined using transmission electron microscope (TEM), one in the as welded state and had never been loaded, while the other was subjected to 300 MPa loading up to 10,000 cycles. The TEM samples were taken from the middle of the testing piece, as close to the weld as possible, in the vicinity of the point marked ‘‘middle’’ shown in Figure1. Unfortunately, no evidence of dislocations was observed in either sample in this very limited examination. This is not to say that dislocations are not present in the original material. However, owing to the small size of the TEM specimen (3 mm in diameter), it is entirely possible that most of the constrains were removed, and residual stress has been relaxed.[45] A more extensive and detailed analysis, outside the scope of the current study, would be required to support the observations of References 12,

44.

C. Residual Stress Relaxation Model

Phenomenological models were often used to study the residual stress relaxation behaviors[27,29–31]since it is extremely diﬃcult to apply the ﬁrst principles in this case. As discussed above, residual stress relaxation is closely related to dislocation movement. Valluri[32] suggested the situation of stress relaxation is similar to creep. Both cause the evolution of stress/strain over a period of time under external loading. The creep controlled by dislocation movement can be expressed by

d

dt¼ Aðr rthÞ

n

eRTQ ½3

where Q (activation energy for dislocation motion), R (universal constant), and T (temperature) are all constants in the case of residual stress; and rth is a

constant dependent on material. Equation [3] can be rewritten as d dt¼ B r rth 1 n ½4

In a manner analogous to Eq. [4], a phenomenological model to predict the amount of residual stress relaxation (S) is proposed: dS dfðNÞ¼ g ra ry ¼ a ra ry n þb ½5

where f(N), which is a function of number of cycles, controls the relaxation rate, while the term g() determines

the amplitude of relaxation. a, b, and n are all material constants. Both Eqs. [4] and [5] are rate-dependent models, i.e., Eq. [4] expresses strain rate, while Eq. [5] gives relaxation rate. The right-hand side of both equations gives the dependence of the rate under con-sideration of existing stress conditions and material parameters. This model was derived by combining the material constants with experimental data. For the mathematical convenience, the expression f(N) is deter-mined to be in the form:

f Nð Þ ¼ ½logðN þ 1Þm ½6 where m is a material constant, and N is the number of loading cycles. Then Eq. [5] can be rewritten in the following form: S¼ a ra ry n þb ½logðN þ 1Þm ½7

In this expression, ra= applied stress and ry= yield

stress. The material constants a, b, m, and n are determined by nonlinear fitting according to the exper-imental data. The relaxation model reveals the depen-dency on applied stress and number of cycles. The amount of relaxation was plotted as a function of cycles in Figure10. The proposed model and showed a very good fitting for both the AISI 1008 and ASTM 572 steels. The fitted value of the material constant, m, is similar under all applied load amplitudes for both materials (m 0.1), which indicates that the amount of relaxation stabilizes at a high rate in all specimens. Figure11 showed the effect of both applied stress amplitude and number of cycles on stress relaxation of AISI 1008 steel. It is obvious that the higher the applied stress, and the larger the number of cycles, the greater the amount of relaxation. The fitting parameters are listed in TableIII. The goodness of fitting (R2) is 0.93.

In order to further test the relaxation model, it was applied to A. Wick’s[14]residual stress relaxation data of AISI 4140 plates. They measured the change of residual stress when applying the cyclic loading on shot-peened AISI 4140 steels. The result is shown in Figure12 and TableIII, and the relaxation model again shows a very good ﬁt with a goodness of ﬁt (R2) equal to 0.92.

Based on the externally applied stress, the residual stress relaxation behavior can be separated into three regimes, as shown in Figure13. These regimes might be considered analogous to the three stages seen in creep, which is not surprising since the model assumes relax-ation similar to that occurs in creep. Behavior in Regime I is not clear since it was not studied, and no data exist. Presumably this might correspond to very low loads where any relaxation would only occur after an exces-sively large number of cycles.

In Regime II, the stress amplitude is intermediate. The amount of relaxation in this regime showed a weak correlation with the applied stress, although it is still determined by Eq. [7]. Instead, the relaxation highly depends on the material constant b, which is a ‘‘thresh-old value’’ similar to rth in Eq. [3] involving creep

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activate the stress relaxation. In the current study, AISI 1008 steel has a higher value of constant b, which indicates that it has more stress relaxation than the

ASTM A572 and A. Wick’s AISI 4140 material under the intermediate loading. It appears the materials with lower yield strength are easier to activate stress relax-ation and thus have higher values of constant b. This conclusion is consistent with observation made by Morrow et al.[44] who found that decay of mean stress is more pronounced for softer materials when they conducted a study on martensitic steels as a function of heat treatment.

In Regime III, where the applied load approximates the yield point of the material, the stress relaxation is proportional to the applied stress and is dominated by the power law (Eq. [7]). The largest amount of relaxa-tion is expected in this regime, although more experi-ments are needed to completely elucidate the exact

0 0.3 1 2 3 4 0.25 0.5 0.75 1 1.25 1.5 0 10 20 30 40 50 log(N+1) Ap plie_{d S} tres_{s (} Nor ma lized) A m ount of S tr e s s R e la x a ti on ( % ) 0 5 10 15 20 25 30 35 40 R2_{= 0.93}

Fig. 11—Eﬀects of both applied stress amplitude and number of cy-cles on stress relaxation of AISI 1008 steel. The solid symbols are experimental data, and the applied stress is normalized to yield strength.

Table III. Fitting Parameters for AISI 1008 steel and A Wick’s[14]_{AISI 4140 steel}

m n a b R2 AISI 1008 0.1 8.0 1.15 0.098 0.93 AISI 4140 0.1 3.0 1.03 0.059 0.92 0 1 2 3 4 5 0 0.25 0.5 0.75 1 1.25 1.5 0 20 40 60 80 100 log(N+1) Ap plie_{d St} res_{s (} No rm aliz ed) A m o unt of S tr e s s R e la x a ti on (% ) 0 10 20 30 40 50 60 70 80 90 100 R2_{= 0.92}

Fig. 12—The application of the proposed relaxation model on data from Ref. [14] The solid symbols are experimental data.

Applied Stress

Amount of Stress Relaxation

Regime II Intermediate Stress b Regime III Power Law Regime I Low Stress

Fig. 13—Residual stress relaxation can be separated into three regimes as a function of applied stress.

0 0.3 1 2 3 (a) (b) 4 5 0 4 8 12 16 18 log(N+1)

Amount of Stress Relaxation (%)

AISI 1008 63MPa 167MPa 250MPa 300MPa g(0.25)=10.24 g(1)=9.72 g(1.2)=15.07 _{g(0.67)=10.46} 0 0.3 1 2 3 4 0 5 10 15 20 25 log(N+1)

Amount of Stress Relaxation (%)

ASTM A572 6mm 100MPa 4mm 400MPa 4mm further 400MPa 6mm 400MPa 6mm further 400MPa g(0.29)=7.1 g(1.16)=21.22 g(1.16)=20.11 g(1.16)=18.51 g(1.16)=18.82

(11)

nature of the relationship. A greater n value will result in a sharper curve between Regime II and Regime III, which means larger discrepancy between these two regimes. It appears that the soft materials like AISI 1008 have the greater n value, indicating faster decay under high stresses.

IV. CONCLUSIONS

The residual stress relaxation behaviors of three materials, AISI 1008, ASTM A572, and AISI 4142, subjected to the cyclic tensile stress at R = 0.1, were documented in the current study. Relaxation occurs at each applied stress in AISI 1008 steel, while no significant relaxation happened in AISI 4142 steel regardless of the applied stress level. ASTM A572 steel showed a conventional relaxation behavior where stress redistribution only starts when the applied stress on the material reaches its yield point. However, all three materials shared some common features, namely that, if stress relaxation was to occur, it was generally com-pleted during the first few loading cycles, and that no significant reduction occurred in the longitudinal stress which is perpendicular to the loading direction.

Stress relaxation is most readily activated in AISI 1008 steel, presumably because of a microstructure consisting of large ferrite grains. The very ﬁne pearlite microstruc-ture with small precipitates seen in AISI 4142 steel can be expected to obstruct dislocation motion and appears to hinder residual stress relaxation. A model based on the principle of creep dislocation movement was proposed to estimate the relaxation of welding residual stress. This model shows good agreement with the experimental data and suggested three regimes based on the applied stresses. The stress relaxation has a weak correlation with inter-mediate applied loads (Regime II), while a power law dependence exists when external stresses over the yield limit of material (Regime III) are applied.

ACKNOWLEDGMENTS

The research was carried out at Iowa State Univer-sity (ISU) with ﬁnancial support granted by Deere and Company. The author would like to thank Jerry Amenson and Scott Schlorholtz of the Materials Anal-ysis and Research Laboratory of ISU, and David Eisenmann of CNDE for their help and permitting access to equipment.

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