Low-Cycle Fatigue Life - Plastic Strain

Coffin-Manson

Coffin-Manson Strain-Life Method

Problem Statement

A linear elastic analysis gives the stress at a point, for example K_t times the nominal stress at a notch root or the peak stress of a linear finite element model. When that stress exceeds the yield strength the material at the point does not carry it: it yields, the stress redistributes to the surrounding elastic material, and the point sees a lower stress and a larger strain than the elastic analysis reports. Fatigue cracks start at such points, and in the low-cycle regime, roughly below 10⁴ cycles, the plastic strain at the point governs the life. The strain-life method therefore works in two steps: convert the elastic stresses of one load cycle into the local elastic-plastic stress and strain, then enter a strain-life relation fitted to strain-controlled tests of smooth specimens. The tool takes the minimum and maximum elastic stresses of the cycle, the cyclic stress-strain constants and the strain-life constants, and returns the local stresses and strains, the Coffin-Manson life, and the two mean-stress corrected lives described on the other pages.

Local Stress and Strain: Neuber's Rule

Neuber's rule states that the product of the local stress and the local strain equals the product of the elastic stress and the elastic strain at the same point: σϵ = σ_1,max²/E. The right-hand side is known from the elastic analysis; the left-hand side is a hyperbola in the stress-strain plane. The local state is the point where that hyperbola meets the stress-strain curve of the material. For cyclic loading the curve is the cyclic stress-strain curve, the locus of the tips of stabilized hysteresis loops, written in Ramberg-Osgood form as

ϵ = σ/E + (σ/K′)^1/n′

with the cyclic strength coefficient K′ (also written H′) and the cyclic strain hardening exponent n′. The tool solves this for the peak of the cycle by Newton's method and shows every iteration. The unloading from the peak follows the hysteresis branch, which by Masing's hypothesis is the cyclic curve with stress and strain doubled: Δϵ = Δσ/E + 2(Δσ/2K′)^1/n′. Neuber's rule applied to the range, ΔσΔϵ = Δσ_e²/E with Δσ_e the elastic stress range, fixes the local stress range and strain range. The local minimum is the peak minus the range, and the strain amplitude, stress amplitude, mean stress and stress ratio R = σ_min/σ_max follow. Figure 1 on the results page draws both constructions. A cycle from zero to a tensile elastic stress usually produces a compressive local minimum, because the peak has yielded and the unloading is elastic-plastic along the doubled curve; the local R is then well below the elastic R of zero.

Neuber's rule is the standard choice and is somewhat conservative for notches under plane stress; the equivalent strain energy density method of Molski and Glinka gives a lower local strain and is not implemented here. The rule assumes the elastic stress is a true peak stress at a free surface, uniaxial, and that the material is the same at the point as in the smooth-specimen tests.

The Strain-Life Relation

Strain-controlled tests at zero mean strain show that the elastic and plastic parts of the strain amplitude each follow a power of the number of reversals 2N_f. Basquin's line for the elastic part and the Coffin-Manson line for the plastic part add to the total:

ϵ_a = (σ′_f/E)(2N_f)^b + ϵ′_f(2N_f)^c

The four constants are fatigue properties of the material. σ′_f, the fatigue strength coefficient, is the stress intercept of the elastic line at one reversal; it is often close to the true fracture strength but is a fitted constant, not the fracture stress. b, the fatigue strength exponent, is the slope of that line, typically −0.05 to −0.15. ϵ′_f, the fatigue ductility coefficient, is the strain intercept of the plastic line at one reversal, and c, the fatigue ductility exponent, its slope, typically −0.5 to −0.8. The two lines cross at the transition life; below it plastic strain dominates and ductility controls, above it elastic strain dominates and strength controls. Figure 2 on the results page draws the lines, their sum, and the solution. The tool solves the relation for 2N_f by Newton's method on ln(2N_f), starting from the elastic-line value where the function is monotonic, and shows the iterations. The result is the life in cycles to crack initiation at the point under constant-amplitude cycling between the two entered stresses.

The Coffin-Manson relation holds for fully reversed loading, R = −1 at the point. It takes no account of the mean stress of the local cycle, so for a tensile local mean it overestimates the life; the Smith-Watson-Topper and Walker pages describe the two corrections the tool applies.

Compatibility of the Cyclic Curve With the Strain-Life Constants

If the Masing loop and the two strain-life lines describe the same material, the cyclic constants follow from the strain-life constants: n′ = b/c and K′ = σ′_f/ϵ′_f^n′. Published sets are fitted separately and rarely satisfy this exactly. The results page computes the compatible values and the ratio of the cyclic-curve stress amplitude to the one implied by the strain-life elastic line at the same strain amplitude, and warns when the entered constants deviate by more than 10 percent. The matter affects the Smith-Watson-Topper life, whose left-hand side takes the local peak stress from the cyclic curve while its right-hand side implies the strain-life stress; it does not affect Coffin-Manson or Walker, which use only the strain amplitude and R.

The Material Entries

The Material dropdown fills the seven constants from fits of public-domain strain-controlled test tables: Endo and Morrow's 1966 Naval Air Engineering Center report (7075-T6, 2024-T4, 4340 at 180 ksi, Ti-8Al-1Mo-1V), NASA TN D-1574 of 1963 (ten steel, titanium and aluminum conditions) and NASA TN D-7532 of 1974 (ten metals including Inconel 718 and 18Ni maraging 300). Each entry is a least-squares fit of the elastic line, the plastic line and the cyclic curve on the same specimens, kept only when it reproduces the measured lives within 0.3 decades rms; the labels carry the source's ultimate strength, and the tooltip gives each source's tested life range. The reports, the transcribed data and the fit records are in the site's References folder. These are 1960s and 1970s heats tested on round bar; check against the current specification before use. The fits for the aluminum alloys are the weakest because their plastic lines are concave downward, as the source authors themselves noted; use those entries within about 10 to 1,000 cycles. The entries cannot set the Walker exponent, since fully reversed tests carry no information about mean-stress sensitivity; see the Walker page.

Limitations

The life is to crack initiation at a point under constant-amplitude cycling in room-temperature air. No factor is applied for the stress gradient at a notch, surface finish, size, temperature, environment or variable-amplitude loading. The method assumes a uniaxial local stress and a material that stabilizes to a single cyclic curve; it is not valid for materials that transform or harden continuously, which is why several conditions of the NASA data were rejected from the dropdown. For lives beyond about 10⁵ cycles the fitted elastic lines of some entries, especially those fitted only to short lives, extrapolate; the stress-life approach and the fatigue limit then govern.

Smith-Watson-Topper SWT

Smith-Watson-Topper Mean-Stress Correction

Problem Statement

The Coffin-Manson relation is fitted to fully reversed tests, where the local stress cycles symmetrically about zero. A load cycle between two elastic stresses that are not equal and opposite leaves the point with a mean stress after yielding, and a tensile mean stress shortens the life. Smith, Watson and Topper proposed in 1970 that the quantity governing fatigue damage is the product of the maximum stress and the strain amplitude of the local cycle, σ_maxϵ_a. Two cycles with the same product are expected to give the same life whatever their mean stress. The quantity has the units of energy per volume and is often called the SWT parameter.

The Equation

For a fully reversed cycle σ_max equals the stress amplitude σ_a = σ′_f(2N_f)^b, and multiplying the Coffin-Manson relation by it gives the SWT parameter as a function of life:

σ_maxϵ_a = (σ′_f²/E)(2N_f)^2b + σ′_fϵ′_f(2N_f)^b+c

The right-hand side needs only the four Coffin-Manson constants; no new material property is introduced. The tool evaluates the left-hand side with the local peak stress and strain amplitude from the Neuber constructions, solves the right-hand side for 2N_f by Newton's method on ln(2N_f), and shows the iterations. On the results page Figure 2 draws the equation divided by the local σ_max as a curve in the strain-life plane, so the crossing with the computed strain amplitude is the SWT life. With a tensile local mean stress the curve lies below the Coffin-Manson curve and the life is shorter; with a compressive mean it lies above.

When the Method Applies

The parameter is meaningful only when the local peak stress is tensile. If σ_max is zero or negative the product is zero or negative and the equation predicts no damage, which is the method's way of saying that a cycle held entirely in compression does not open a crack. The tool then omits the SWT and Walker results and reports the Coffin-Manson life only. In practice the method does well for aluminum alloys and for steels at moderate mean stresses; at high tensile means it tends to be unconservative for some steels, which is one motive for the Walker exponent.

Relation to the Walker Equation

In stress-life terms the SWT parameter is the Walker equation with γ = 0.5: the equivalent fully reversed stress amplitude is the geometric mean of the maximum stress and the amplitude, σ_ar = (σ_maxσ_a)^1/2. Dowling's strain-life form of the Walker equation at γ = 0.5 is therefore often called the strain-based SWT. It is not the same equation as the one above. The classic SWT form multiplies the whole strain amplitude by σ_max, which scales the product by the single factor 2/(1 − R); the Walker strain-life form replaces 2N_f by an equivalent number of reversals in both terms, which scales the plastic term by a larger power. In the low-cycle regime, where the plastic term dominates, the two differ, and the Walker life at γ = 0.5 is shorter than the classic SWT life. The tool solves the classic form in this section and the Walker form in the next, and reports both.

Sensitivity to the Cyclic Curve

Unlike the other two methods, SWT uses the local peak stress, which comes from the entered cyclic stress-strain curve. Its right-hand side implies the stress of the strain-life elastic line at the same strain amplitude. When the cyclic constants are not compatible with the strain-life constants, the ratio of those two stresses multiplies the mean-stress factor and can mask or exaggerate it; with some published constant sets the SWT life lands next to the Coffin-Manson life even with a sizable tensile mean. The results page prints the compatible cyclic constants and that ratio under the Coffin-Manson section so the effect is visible. Entries from the Material dropdown have cyclic and strain-life constants fitted to the same specimens and are free of it.

Walker

Walker Mean-Stress Correction

Problem Statement

Materials differ in how much a tensile mean stress shortens their fatigue life. The Goodman, Morrow and Smith-Watson-Topper corrections each fix that sensitivity by a formula, with no adjustable constant, so none of them can fit a material that is more or less sensitive than the formula assumes. Walker proposed in 1970 an equivalent fully reversed stress amplitude that weights the maximum stress and the amplitude with an exponent γ fitted to the material:

σ_ar = σ_max^1−γσ_a^γ = σ_a((1 − R)/2)^γ−1

with R = σ_min/σ_max. At γ = 0.5 this is the SWT geometric mean; at γ = 1 the mean stress has no effect; as γ falls toward zero the maximum stress alone controls. A lower γ therefore means a material more sensitive to mean stress. Dowling, Calhoun and Arcari showed in 2009, on data for eighteen steels, nine aluminum alloys and one titanium alloy, that the Walker form fits mean-stress data better than the fixed formulas and that γ follows the strength of the material.

The Strain-Life Form

Dowling carried the equation into the strain-life relation in 2004 by requiring the stress amplitude of the Basquin line to equal the equivalent fully reversed amplitude. Writing σ_a = σ′_f((1 − R)/2)^1−γ(2N_f)^b and defining an equivalent number of reversals 2N* = 2N_f((1 − R)/2)^(1−γ)/b, the Coffin-Manson relation holds in terms of N*:

ϵ_a = (σ′_f/E)(2N*)^b + ϵ′_f(2N*)^c

The tool writes the same equation with the factor f_W = ((1 − R)/2)^1−γ on the elastic term and f_W^c/b on the plastic term, which is what the substitution of N* produces, and solves it for 2N_f by Newton's method on ln(2N_f). At R = −1 the factor is 1 and the Coffin-Manson life is recovered. Because c/b is typically 4 to 8, a modest reduction of the elastic term becomes a large reduction of the plastic term, so in the low-cycle regime the Walker life is markedly shorter than the Coffin-Manson life even for a moderate tensile mean, and shorter than the classic SWT life, which scales the product by 2/(1 − R) only. Figure 2 on the results page draws the Walker curve for the computed R. The form was checked against its reproduction as Equation 14 of Ince and Glinka's 2011 paper, filed in the site's References folder.

R here is the local stress ratio from the Neuber constructions, not the ratio of the elastic stresses; for a cycle from zero to a yielding tensile peak it is typically −0.5 to −0.8. The method requires a tensile local peak: for σ_max ≤ 0 the ratio is 1 or undefined and the factor has no meaning, so the tool omits the Walker and SWT results and reports Coffin-Manson only.

Values of the Exponent

γ cannot be determined from fully reversed tests; it needs tests at two or more mean stresses. The published fits are those of Dowling, Calhoun and Arcari (2009). For carbon, low-alloy and other hardenable steels γ falls linearly with ultimate tensile strength, γ = 0.8818 − 0.000200 σ_u with σ_u in MPa, which gives about 0.72 at 800 MPa and 0.48 at 2000 MPa. High-strength aluminum alloys of the 2014, 2024 and 7075 class have γ of 0.41 to 0.57 with a mean near 0.47, close to the SWT value of 0.5; lower-strength aluminum alloys with ultimate strengths of 300 to 350 MPa have about 0.65. No published fit exists for austenitic stainless steels, titanium alloys, nickel alloys or copper. The Material dropdown sets γ from the steel line and the aluminum class value for the entries those fits cover, and clears the field for the others so that a value must be chosen deliberately; 0.5 is the usual starting point when nothing is known. The sources behind these numbers are listed on the references page and filed in the References folder; the 2009 paper itself is behind a paywall and was not reproduced.

Reading the Three Lives

With a tensile local mean the results page normally shows the Coffin-Manson life longest, the classic SWT life next and the Walker life shortest, the last two converging as γ approaches 1 and crossing only when γ is small. For a fully reversed local cycle all three coincide. For design the Walker life with a fitted γ is the best supported of the three; where γ is a guess, the spread between the SWT and Walker results is a fair measure of the uncertainty from mean-stress sensitivity alone.