Following the recent post on the Dang Van criterion, this companion post covers its two closest relatives: the Crossland criterion (1956) and the Sines criterion (1959). A quick note on nomenclature: the Crossland criterion is sometimes misremembered as “Crossfield” — the correct attribution is to Bernard Crossland, whose 1956 paper on the effect of hydrostatic pressure on torsional fatigue launched this family of methods. Together with Dang Van, these invariant-based criteria form the standard toolkit for infinite-life screening of components under combined bending, torsion, pressure, and residual stress.
1. The Invariant-Based Family
Multiaxial high-cycle fatigue criteria fall broadly into three camps: critical plane methods (Findley, Matake, McDiarmid, Papuga), which search over candidate planes for the worst combination of shear and normal stress; mesoscopic methods (Dang Van), which model grain-scale shakedown; and invariant-based methods, which condense the entire stress tensor history into scalar invariants. Crossland and Sines are the founding members of the third camp, and their appeal is unbeatable simplicity: two numbers per load cycle, two material constants, done.
Both criteria share the same architecture. Fatigue damage is driven by an octahedral (von Mises-type) shear stress amplitude, penalized by a hydrostatic stress term that captures mean stress sensitivity:
\[ \sqrt{J_{2,a}} + \alpha \, \sigma_{H}^{*} \; \leq \; \beta \]
They differ in exactly one design decision: which hydrostatic quantity \( \sigma_H^{*} \) to use. Crossland uses the maximum hydrostatic stress over the cycle; Sines uses the mean. That single choice turns out to have large consequences.
2. The Ingredients
Hydrostatic stress. The instantaneous hydrostatic stress is one-third the trace of the stress tensor:
\[ \sigma_H(t) = \tfrac{1}{3} \left[ \sigma_{xx}(t) + \sigma_{yy}(t) + \sigma_{zz}(t) \right] \]
From its history over the cycle, take either the maximum \( \sigma_{H,max} \) (Crossland) or the mean \( \sigma_{H,m} \) (Sines).
Deviatoric shear amplitude. \( J_2 \) is the second invariant of the deviatoric stress tensor; \( \sqrt{J_2} \) is proportional to the octahedral shear stress and equals \( \sigma_{vm}/\sqrt{3} \) for the instantaneous von Mises stress. The amplitude \( \sqrt{J_{2,a}} \) over a general multiaxial cycle is defined via the minimum circumscribed hypersphere of the deviatoric stress path in five-dimensional deviatoric stress space — the same geometric construction used in Dang Van’s method. The hypersphere radius is the amplitude; its center represents the deviatoric mean, which both criteria discard as non-damaging (consistent with the experimental observation that mean torsional stress has little effect on the torsional fatigue limit, whereas mean tensile stress matters greatly).
For proportional loading the hypersphere construction collapses to the obvious result: \( \sqrt{J_{2,a}} = \sigma_{vm,a}/\sqrt{3} \), the von Mises amplitude over root three. For fully reversed uniaxial stress of amplitude \( \sigma_a \), \( \sqrt{J_{2,a}} = \sigma_a/\sqrt{3} \); for fully reversed torsion of amplitude \( \tau_a \), \( \sqrt{J_{2,a}} = \tau_a \).
3. The Crossland Criterion
\[ \sqrt{J_{2,a}} + \alpha_C \, \sigma_{H,max} \; \leq \; \beta_C \]
Calibration from the fully reversed bending (or axial) fatigue limit \( f_{-1} \) and the fully reversed torsion fatigue limit \( t_{-1} \):
\[ \alpha_C = \frac{3\, t_{-1}}{f_{-1}} – \sqrt{3}, \qquad \beta_C = t_{-1} \]
The torsion test fixes \( \beta_C \) directly, since fully reversed torsion has zero hydrostatic stress at all times. The bending test then fixes \( \alpha_C \). Note the applicability condition: \( \alpha_C > 0 \) requires \( t_{-1}/f_{-1} > 1/\sqrt{3} \approx 0.577 \). Ductile metals typically show ratios of about 0.55 to 0.65, so some perfectly ordinary materials sit uncomfortably close to (or below) Crossland’s floor — a point we will see vividly in the worked example.
4. The Sines Criterion
\[ \sqrt{J_{2,a}} + \alpha_S \, \sigma_{H,m} \; \leq \; \beta_S \]
Sines’ use of the mean hydrostatic stress creates an immediate structural quirk: fully reversed bending has zero mean hydrostatic stress, so the bending fatigue limit cannot be used to calibrate \( \alpha_S \). Worse, the criterion is forced to predict a fixed relationship between the two fully reversed limits:
\[ f_{-1} = \sqrt{3}\; t_{-1} \]
regardless of the material constants — a prediction the material is under no obligation to obey. Calibration therefore requires a test with nonzero mean stress, typically the pulsating (R = 0) axial fatigue limit \( f_0 \) (maximum stress at the limit, amplitude \( f_0/2 \), mean \( f_0/2 \)). With the torsion limit setting \( \beta_S = t_{-1} \), the R = 0 test gives:
\[ \alpha_S = \frac{6\, t_{-1}}{f_0} – \sqrt{3} \]
The compensating virtue: because only the mean hydrostatic stress enters, the Sines criterion is extremely cheap to evaluate and completely insensitive to how the load components are phased within the cycle — which is either a feature (robust, simple) or a bug (blind to non-proportionality), depending on the application.
5. Worked Example
Take the same steel used in the Dang Van post: \( f_{-1} = 60 \) ksi, \( t_{-1} = 35 \) ksi, so \( t_{-1}/f_{-1} = 0.583 \) — just above Crossland’s floor of 0.577.
Crossland constants.
\[ \alpha_C = \frac{3(35)}{60} – \sqrt{3} = 1.750 – 1.732 = 0.018, \qquad \beta_C = 35 \; \text{ksi} \]
Consistency check. Fully reversed bending at amplitude \( \sigma_a \): \( \sqrt{J_{2,a}} = \sigma_a/\sqrt{3} \) and \( \sigma_{H,max} = \sigma_a/3 \). The criterion reads \( \sigma_a (0.5774 + 0.006) \leq 35 \), giving \( \sigma_a \leq 60 \) ksi. The bending limit is recovered exactly, as it must be.
Mean stress sensitivity — and a surprise. Superimpose a static tension of 30 ksi, so \( \sigma_{H,max} = 10 + \sigma_a/3 \):
\[ \frac{\sigma_a}{\sqrt{3}} + 0.018\left( 10 + \frac{\sigma_a}{3} \right) \leq 35 \;\;\Rightarrow\;\; \sigma_a \leq 59.7 \; \text{ksi} \]
Crossland knocks the allowable amplitude down by only 0.3 ksi. Recall that Dang Van, calibrated to the same two fatigue limits, predicted 55.7 ksi — a fourteen-fold larger knockdown. The culprit is the tiny \( \alpha_C \): because this material’s \( t_{-1}/f_{-1} \) ratio sits barely above \( 1/\sqrt{3} \), Crossland concludes it is nearly insensitive to hydrostatic stress. Dang Van, whose calibration formula has a different structure (its floor is at \( t_{-1}/f_{-1} = 0.5 \)), retains healthy mean stress sensitivity for the same inputs. Same data, same family of methods, very different engineering conclusion — a caution worth remembering whenever a criterion is used near the edge of its applicability range.
A second material for contrast. Take \( f_{-1} = 60 \) ksi, \( t_{-1} = 39 \) ksi (ratio 0.65). Then \( \alpha_C = 1.95 – 1.732 = 0.218 \) and \( \beta_C = 39 \) ksi. The same 30 ksi static tension case gives \( \sigma_a (0.5774 + 0.0727) \leq 39 – 2.18 \), or \( \sigma_a \leq 56.6 \) ksi versus 60 ksi with no mean stress — a substantial and physically sensible knockdown. Crossland behaves well when the material ratio gives it room to work.
Sines on the first material. \( \sqrt{3} \times 35 = 60.6 \) ksi versus the actual \( f_{-1} = 60 \) ksi — for this particular steel, the forced Sines relationship happens to land within one percent, which is exactly why Sines survived decades of use: for many ductile steels the ratio sits near \( 1/\sqrt{3} \) and the structural defect is invisible. For the second material, Sines would predict \( f_{-1} = \sqrt{3} \times 39 = 67.5 \) ksi against an actual 60 ksi — a 12% unconservative error on the most basic test in the book, before any multiaxiality enters.
6. Side-by-Side Comparison
| Feature | Sines (1959) | Crossland (1956) | Dang Van (1973+) |
|---|---|---|---|
| Shear measure | \( \sqrt{J_{2,a}} \) (octahedral) | \( \sqrt{J_{2,a}} \) (octahedral) | Mesoscopic max shear (Tresca-type) |
| Hydrostatic term | Mean over cycle | Maximum over cycle | Instantaneous, at every time step |
| Time resolution | Cycle-level scalars | Cycle-level scalars | Point-by-point in time |
| Calibration tests | Reversed torsion + pulsating (R=0) axial | Reversed torsion + reversed bending | Reversed torsion + reversed bending |
| Applicability floor | Forces \( f_{-1} = \sqrt{3}\, t_{-1} \) | \( t_{-1}/f_{-1} > 0.577 \) | \( t_{-1}/f_{-1} > 0.5 \) |
| Non-proportional loading | Blind to phasing | Partially sensitive (via hypersphere and \( \sigma_{H,max} \)) | Sensitive (instantaneous check) |
| Computational cost | Lowest | Low | Moderate |
7. Which One to Use?
For quick screening of proportional or nearly proportional loading with materials whose \( t_{-1}/f_{-1} \) ratio sits comfortably in the 0.6–0.7 range, Crossland is hard to beat: two standard fatigue limits, one hypersphere, one inequality. It remains a favorite in additive manufacturing fatigue studies, gear and bearing work, and pressure-superimposed applications — fitting, given that Crossland’s original experiments concerned torsional fatigue under hydrostatic pressure. Sines is best regarded as a historical stepping stone and a pedagogical example: its mean-hydrostatic choice makes it structurally unable to fit both fully reversed limits, and modern comparative studies (notably the large Papuga database validations) consistently rank it below Crossland and Dang Van. For strongly non-proportional loading — rotating principal directions, out-of-phase bending and torsion — the whole invariant family loses accuracy, and Dang Van or a critical plane method (Findley, Papuga) is the better tool. Near the applicability floors, as the worked example showed, small changes in the input fatigue limits swing the predicted mean stress sensitivity wildly, so treat results there with appropriate suspicion and test data.
8. Connection to Vibration Fatigue
As with Dang Van, these are infinite-life criteria: they answer whether a crack initiates, not how fast damage accumulates under random loading. Their scalar structure, however, makes them attractive building blocks for multiaxial spectral fatigue: an equivalent von Mises-type PSD can be constructed from the stress-component PSD matrix, reducing the multiaxial random problem to an equivalent uniaxial one to which Dirlik-family methods — including corrections such as Meta-Dirlik — can be applied, with a hydrostatic penalty layered on in the spirit of Crossland. The interplay between invariant-based screening and spectral damage estimation is an active research area and a likely subject for a future post.
9. Closing
Crossland and Sines bracket a single design decision — peak versus mean hydrostatic stress — and the sixty-plus years of validation data since have rendered a clear verdict in Crossland’s favor. Together with Dang Van, they illustrate a recurring theme in fatigue analysis: the criteria that survive are the ones that are cheap to calibrate, cheap to compute, and honest about their applicability limits. Know the floors, check your material’s \( t_{-1}/f_{-1} \) ratio before trusting the constants, and keep a critical plane method in reserve for the strongly non-proportional cases.
I cover fatigue analysis methods, including rainflow cycle counting, spectral fatigue, and mean stress corrections, in my continuing education course series beginning in September 2026, with IACET-accredited CEUs available. Details are at vibrationdata.com.
References
B. Crossland, “Effect of Large Hydrostatic Pressures on the Torsional Fatigue Strength of an Alloy Steel,” Proceedings of the International Conference on Fatigue of Metals, Institution of Mechanical Engineers, London, 1956.
G. Sines, “Behavior of Metals Under Complex Static and Alternating Stresses,” in Metal Fatigue, G. Sines and J.L. Waisman (eds.), McGraw-Hill, 1959.
K. Dang Van, B. Griveau, and O. Message, “On a New Multiaxial Fatigue Limit Criterion: Theory and Application,” in Biaxial and Multiaxial Fatigue, EGF 3, Mechanical Engineering Publications, London, 1989.
I.V. Papadopoulos, P. Davoli, C. Gorla, M. Filippini, and A. Bernasconi, “A Comparative Study of Multiaxial High-Cycle Fatigue Criteria for Metals,” International Journal of Fatigue, Vol. 19, No. 3, 1997.
J. Papuga, “A Survey on Evaluating the Fatigue Limit under Multiaxial Loading,” International Journal of Fatigue, Vol. 33, 2011.
D.F. Socie and G.B. Marquis, Multiaxial Fatigue, SAE International, 2000.