Most of the fatigue methods featured on this blog — rainflow counting, Dirlik and my Meta-Dirlik spectral correction, dwell-corrected rainflow — answer the question: how much damage does a stress history accumulate? The Dang Van criterion answers a different and complementary question: will a complex multiaxial stress state ever initiate a fatigue crack at all? It is the workhorse infinite-life criterion of the European automotive and rail industries, and it deserves a place in every structural dynamicist’s toolbox. This post covers its physical basis, its equations, a worked example, and how it connects to the vibration fatigue methods regularly covered here.
1. Background
The criterion is named for Ky Dang Van, who developed it in France beginning in the 1970s, with the mature form appearing in the 1980s and 1990s through his work at École Polytechnique and his collaborations with the French automotive and railway industries. It was created to solve a practical problem: components such as crankshafts, rail wheels, suspension arms, and gear teeth experience multiaxial stress states — often non-proportional, meaning the principal stress directions rotate during the load cycle — and the classical uniaxial fatigue limit simply does not apply.
Two observations doomed the naive approaches. First, an equivalent-stress criterion built on von Mises stress cannot work by itself, because von Mises stress is insensitive to hydrostatic stress, while fatigue limits demonstrably depend on mean and hydrostatic stress — this is the entire content of the Goodman and Gerber corrections in the uniaxial world. Second, criteria built purely on macroscopic stress amplitudes fare poorly under non-proportional loading, where the damaging shear can occur on planes that a proportional analysis never interrogates.
2. The Mesoscopic Scale Concept
Dang Van’s central insight is one of scale. High-cycle fatigue cracks initiate from plastic slip in individual, unfavorably oriented metal grains — even when the component’s bulk stress state is nominally elastic and well below yield. The relevant physics therefore lives at the mesoscopic (grain) scale, not the macroscopic (engineering stress) scale.
During the first cycles of loading, these plastically active grains work-harden and develop local residual stresses. If the loading is at or below the fatigue limit, the grains reach elastic shakedown: a stabilized state in which the local residual stress field has adjusted so that all subsequent response is elastic. Dang Van postulates that a crack will not initiate provided the shaken-down mesoscopic stress state remains safe at every instant of the stabilized cycle — not merely at the peaks.
The mesoscopic stress is related to the macroscopic stress through the stabilized local residual stress tensor \( \rho^* \):
\[ \hat{\sigma}(t) = \sigma(t) + \rho^* \]
Two convenient facts make the method tractable. The hydrostatic stress is essentially scale-invariant — the same at the grain scale as at the macroscopic scale — and the stabilized residual stress is purely deviatoric. The problem therefore reduces to finding the mesoscopic shear stress history.
3. The Criterion
The Dang Van criterion states that fatigue crack initiation is avoided if, at every time \(t\) in the stabilized load cycle:
\[ \tau(t) + a\, \sigma_H(t) \; \leq \; b \]
where:
\( \tau(t) \) is the instantaneous mesoscopic shear stress — commonly taken as the maximum shear (Tresca-type) of the mesoscopic stress tensor; \( \sigma_H(t) = \tfrac{1}{3}\,\mathrm{tr}\,\sigma(t) \) is the instantaneous hydrostatic stress; and \(a\), \(b\) are material constants.
Finding the mesoscopic shear. The stabilized residual stress \( \rho^* \) is identified as the deviatoric tensor that minimizes the maximum distance to the macroscopic deviatoric stress path over the cycle. Geometrically, one constructs the smallest hypersphere circumscribing the deviatoric stress path in five-dimensional deviatoric stress space; \( \rho^* \) is the negative of the hypersphere center, and the mesoscopic deviatoric stress is the macroscopic path re-centered on that point. The instantaneous maximum shear of the re-centered tensor is \( \tau(t) \). For simple loadings this reduces to intuitive results; for general FEA stress histories it is a small optimization problem solved at each critical location.
Calibrating \(a\) and \(b\). Two fatigue limits suffice. Using the fully reversed bending (or tension) fatigue limit \( f_{-1} \) and the fully reversed torsion fatigue limit \( t_{-1} \):
\[ a = \frac{3 \left( t_{-1} – f_{-1}/2 \right)}{f_{-1}}, \qquad b = t_{-1} \]
The torsion test pins down \(b\) directly because fully reversed torsion produces zero hydrostatic stress. The bending test then fixes the slope \(a\), which expresses the material’s sensitivity to hydrostatic stress — the multiaxial generalization of mean stress sensitivity.
4. Graphical Interpretation
The natural presentation is the Dang Van diagram: plot the loading path parametrically in the \( (\sigma_H, \tau) \) plane over one stabilized cycle, and draw the limit line
\[ \tau = b – a\, \sigma_H \]
If the entire path stays below the line, infinite life is predicted; if the path crosses the line at any instant, crack initiation is predicted. A safety factor follows from the closest approach of the path to the line, commonly defined as
\[ SF = \min_t \; \frac{b}{\tau(t) + a\,\sigma_H(t)} \]
evaluated over the instants where the denominator is positive. This point-by-point-in-time character is what allows the criterion to handle non-proportional loading gracefully: phase shifts between stress components reshape the path in the \( (\sigma_H, \tau) \) plane, and the criterion responds accordingly.
5. A Worked Example
Consider a steel with a fully reversed bending fatigue limit \( f_{-1} = 60 \) ksi and a fully reversed torsion fatigue limit \( t_{-1} = 35 \) ksi. Then:
\[ a = \frac{3(35 – 30)}{60} = 0.25, \qquad b = 35 \; \text{ksi} \]
Note the ratio \( t_{-1}/f_{-1} = 0.583 \), comfortably inside the ~0.5–0.8 range where the criterion is considered applicable to ductile metals.
Check 1 — consistency with the bending limit. For fully reversed uniaxial stress \( \sigma(t) = \sigma_a \sin \omega t \), the maximum shear is \( \sigma_a/2 \) and the hydrostatic stress is \( \sigma_a/3 \), both peaking simultaneously (proportional loading, and the circumscribing construction introduces no offset for a fully reversed path). The criterion at the worst instant reads:
\[ \frac{\sigma_a}{2} + 0.25\,\frac{\sigma_a}{3} \leq 35 \;\;\Rightarrow\;\; 0.5833\, \sigma_a \leq 35 \;\;\Rightarrow\;\; \sigma_a \leq 60 \; \text{ksi} \]
The criterion returns exactly the bending fatigue limit, as it must, since that test calibrated it.
Check 2 — the effect of a static tensile mean stress. Now superimpose a static tension \( \sigma_m = 30 \) ksi on the fully reversed bending. The shear amplitude is unchanged at \( \sigma_a/2 \) (the static term shifts the deviatoric path but the circumscribing hypersphere construction removes a constant deviatoric offset), while the hydrostatic stress now oscillates about a mean: \( \sigma_H(t) = 10 + (\sigma_a/3) \sin\omega t \). At the worst instant:
\[ \frac{\sigma_a}{2} + 0.25\left( 10 + \frac{\sigma_a}{3} \right) \leq 35 \;\;\Rightarrow\;\; 0.5833\,\sigma_a \leq 32.5 \;\;\Rightarrow\;\; \sigma_a \leq 55.7 \; \text{ksi} \]
The allowable amplitude drops from 60 to 55.7 ksi — a mean stress knockdown emerging automatically from the hydrostatic term, with no separate Goodman construction required. This is the criterion’s elegance: mean stress, residual stress, and multiaxiality are all handled by one mechanism.
6. Implementation in Practice
In industrial use, the Dang Van criterion is applied as an FEA post-processing step. A load cycle (or measured service cycle) is applied to the model; at each node or element of interest, the six-component stress history is extracted; the smallest circumscribing hypersphere of the deviatoric path is computed; the mesoscopic shear history and hydrostatic history are formed; and the minimum safety factor over the cycle is reported as a contour plot. Commercial fatigue codes offer it as a standard option, and the hypersphere problem — a minimax optimization — has well-established algorithms. The method is fast enough to run over full-vehicle or full-bogie models, which is a large part of why the automotive and rail industries adopted it.
7. Strengths and Limitations
Strengths. The criterion handles non-proportional multiaxial loading with rotating principal directions; it accounts for mean and residual stresses through the hydrostatic term, which also makes it naturally compatible with shot peening and other residual-stress treatments; it is grounded in a physically motivated shakedown argument rather than curve-fitting; it requires only two standard fatigue limits to calibrate; and it is computationally cheap enough for full-model FEA screening.
Limitations. It is an infinite-life (endurance) criterion — it predicts whether cracks initiate, not how fast damage accumulates, so it does not directly replace rainflow-plus-S-N for finite-life spectrum loading. It can be unconservative for certain out-of-phase load combinations reported in the experimental literature, and its applicability is generally restricted to ductile metals with \( t_{-1}/f_{-1} \) between roughly 0.5 and 0.8 — hard or brittle materials fall outside the shakedown argument. Defect-dominated materials (castings, additively manufactured metals with porosity) are better served by approaches like the Murakami \( \sqrt{\text{area}} \) model, covered previously on this blog. And the criterion addresses initiation only; propagation questions belong to fracture mechanics.
8. Connection to Vibration Fatigue
Readers of this blog will recognize a division of labor. The Dang Van criterion asks: does this multiaxial stress path ever initiate a crack? Rainflow counting and the spectral methods — Dirlik and its corrections — ask: how much damage does this random stress history accumulate per unit time? The two meet in the growing field of multiaxial random vibration fatigue, where researchers project the multiaxial random stress state onto candidate critical planes or equivalent scalar processes and then apply spectral damage estimation plane by plane. A Dang Van-style screening can also identify which locations in a large model deserve the expense of full spectral or time-domain fatigue analysis. Extending the hydrostatic-sensitivity idea into the PSD domain — alongside corrections like Meta-Dirlik — strikes me as fertile ground, and perhaps a future post.
9. Closing
The Dang Van criterion is a rare thing in fatigue analysis: a method that is simultaneously physically principled, cheap to calibrate, cheap to compute, and validated by decades of industrial use on safety-critical hardware. If your components see combined bending, torsion, pressure, and residual stress — and most real hardware does — it belongs in your screening workflow alongside the uniaxial and spectral tools.
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
K. Dang Van, “Sur la résistance à la fatigue des métaux,” Sciences et Techniques de l’Armement, Vol. 47, 1973.
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.
K. Dang Van, “Macro-Micro Approach in High-Cycle Multiaxial Fatigue,” in Advances in Multiaxial Fatigue, ASTM STP 1191, 1993.
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.
D.F. Socie and G.B. Marquis, Multiaxial Fatigue, SAE International, 2000.
Y. Murakami, Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions, 2nd ed., Academic Press, 2019.