Data on Cyclic Mean Stress Relaxation in Mild Steel

3A Civil Engr., April 1970
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(Work Term Report for the Dept. of Co-ordination and Placement, University of Waterloo.)
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The author is indebted to Professor T.H.Topper and H.R.Jhansale for their supervision and guidance throughout the project, and to F.Andrewartha and P.Watson for their stimulating discussions and advice on the subject of fatigue.

List of Symbols


There are three areas, in the Low Cycle Fatigue Research program at the University of Waterloo, where cyclic mean stress behaviour has become relevant:
  1. Micromechanisms: Andrewartha [1] is providing links between the areas of material physics and mechanics by subjugating dislocation theories to macroscopic tests. Mean stress is one of the variables.
  2. Cumulative damage criteria such as those of Smith et al [2] have found that mean stress alters the fatigue life: better predictions of mean stress behaviour will allow improved predictions of life.
  3. Simulations of hysteritic behaviour such as the computer models of Martin et al[3] and the proposal of Jhansale[4], will eventually incorporate the most feasible mechanisms and the damage theory to provide means of predicting full scale structural fatigue response.

The present test series was directly related to the third area; the tests were supervised by H.R.Jhansale, but it is thought that the data presented shall also be of some use to the other two fields of investigation.

Previous work on the same subject for different materials can be found in References [5] and [6]; the test program being partially modeled on the work of Ref.[5].

Testing Program

The material investigated was normalized mild steel. The chemical composition is given in Table 1. Fatigue properties are described by Ref. [7].

Standard longitudinal specimens of 0.375" gage length and 0.250" diameter were tested in an MTS 20 kip closed loop servo controlled system as described in Ref. [8]. Conditioned output signals from load cell and clip - on extensometer were recorded by both X-Y plotter and high frequency response strip chart.

All eight specimens were subjected to strain controlled deformation programs; characterized by a block of cycles of large strain range ( Δε₁ ) followed by a secondary block at a smaller strain range ( Δε₂ ) which "relaxed" the mean stress induced by the chage in strain limits. Typical hysteretic control conditions for various parts of the test are shown in Fig._1 and a sample strain/time and load/time recording is given in Fig. 2.

The test program emphasized mean stress relaxation behviour rather thatn the methods used to induce it. For the normalized mild steel, which displays hardening at high strain rages, and softening in lower nonelastic regions, the relatively stable strain range of Δε₁ = 0.77% was chosen for the "primary cycling block" of six of the specimens, so as to minimize hardening and softening effects on mean stress relaxation. Two other tests, as listed in Table 2 were conducted at the primary range of Δε₁=1.6%.

Secondary strain ranges varied from Δε₂=0.10% to Δε₂=0.67%. In most tests the application of the first two blocks of mean stress inducing ( labeled as 1a for all tests), and means stress relaxing (block type 1b) was followed by others at different ranges. The notation used to denote primary blocks is as:

          Part 1a, 2a, 3a  ...etc.,

while secondary relaxation blocks are labeled:

          Part 1b, 2b, 3b, ...etc.

Mean stress was taken as the average between upper and lower stress limits for every second half cycle and recorded as "mean stress per cycle". Hardening and softening, taken as the change in stress range were also recorded.


A history of all tests in terms of blocks, strain ranges, initial stress range at the start of the block and final stress range at the end of block are given in Table 2. Mean stresses are recorded vs. cycles for all 1b type blocks in Fig. 3, and separately for test blocks of similar strain ranges and different histories in Figs._4 and 7. Stress range vs. cycles for type 1b blocks is given in Fig.8 to give some indication of softening behaviour during mean stress relaxation. Softening for the other blocks can be determined from Table 2.

Figures 4 to 7 indicate that mean stress relaxation as a function of cycles can be expressed as

           So = C x Nb

where  So is the mean stress at cycle N
       C  is the mean stress at cycle 1.
       b  is the slope of So vs. N on the log-log graph.

The exponents for all blocks with mean stress relaxation are plotted vs. total strain range and plastic strain range in Figs. 9 and 10 respectively.

The co-efficient appears to be related to the loop shape of the primary strain limits and the relative poition of the secondary strain limits.


From the test data it can be concluded that mild steel in the "saturated" condition, where relatively liitle hardening or softening occurs, relaxes mean stress under strain cycling according to the empirical equation:
           So = C x Nb

The co-efficient, C is a function of the relative poitiions of primary and secondary strain ranges.

The exponent b, is believed to be related to the plastic strain occurring in the secondary block; as indicated by the trends in Figs._9 and_10. Significant scatter is also evident in these figures; specifically the exponents for tests RX-3 6c, and RX-8_3a. Table 2 indicates that these recordings were made fairly late in the life of the specimens and it is possible that the presence of a crack has affected the behaviour. This however cannot be the only reason for the scatter as other tests at similar lives still conform to the power relationship, without the variation in exponents.

The effects of superimposed hardening or softening behariour on mean stress relaxation are difficult to interpret from present data. By saturation of the material with 500 to 1000 cycles at an intermediate strain range, it was hoped that large changes in stress range with cycling, would not occur. Table 2 indicates however that the mean stresses were at times accompanied by some softening. One possible way for cyclic hardening of softening, defined as a change in stress range, to show up as an apparent mean stress relaxation, is for the material to become anisotropic with deformation. i.e. The stress range in the tensile half cycle is always larger or smaller than the stress range in the compressive half cycle which immediately follows it. This would result in a movement of the hysteresis loop along its mean strain axis with cycling. Definite conclusions as to this inter-relationship can however not be made fro the results of the present study and further research is warranted.


  1. Andrewartha, F., MSc. Thesis, U. of Waterloo, 1970.

  2. Smith, K.N., P.Watson and T.H.Topper, "A Stress-Strain Function for the Fatigue of Metals," Solid Mechanics Division, U.Waterloo, Oct. 1969.

  3. Martin, J.F., T.H.Topper, G.M.Sinclair, "Computer Based Simulation of Cyclic Stress Strain Behavior," T.A.M. Report No. 326, U.Illinois, Urbana

  4. Jhansale, H.R., Ph.D. Thesis, U.Waterloo, 1970.

  5. Topper, T.H., B.I.Sandor, J.Morrow, "Cumulative Fatigue Damage under Cyclic Strain Control," J.Materials, V4 N1 March 1969.

  6. Morrow, J., G.M.Sinclair, "Cycle-Dependent Stress Relaxation," ASTM Symp. on Basic Mechanisms of Fatigue, ASTM Special Tech. Publ. No. 237.

  7. Keshavan, S., "Some Studies on the Deformation and Fracture of Normalized Mild Steel under Cyclic Conditions," PhD Thesis, U.Waterloo, 1967.

  8. Butzow, G.N., R.W.Churchill, "Electrohydraulic Fatigue Test Systems Capabilities," SESA Annual Meeting Cleveland, Ohio, Oct. 28-30, 1964.