Introduction
Shock and vibration environments are almost always characterized in terms of velocity. The shock response spectrum (SRS) is fundamentally a pseudo-velocity construct. Random vibration specifications are integrated to give RMS velocity as a check on response severity. Field data from accelerometers is routinely converted to velocity for machinery condition monitoring, where ISO 10816 and related standards set vibration severity limits in velocity units precisely because velocity correlates well with vibration energy and fatigue damage potential across a wide range of structures.
Fatigue and fracture assessment, on the other hand, is almost always framed in terms of stress, strain, or energy density. This creates a persistent translation problem for anyone working in shock and vibration: the environment is measured and specified in velocity, but the failure criterion lives in stress-strain space.
The classical stress-velocity relationship, $\sigma = \rho c V$, has long served as a fast, physically grounded bridge between these two worlds. This post examines that relationship more closely, shows how it connects naturally to the strain energy density (SED) framework covered in a previous post, and proposes a velocity-based SED parameter that lets an engineer move directly from a measured or predicted response velocity to a fatigue or fracture damage estimate.
The Stress-Velocity Relationship
For a uniaxial stress wave propagating in an elastic, isotropic rod, the relationship between dynamic stress $\sigma$ and particle velocity $V$ is:
$$\sigma = \rho c V$$
where $\rho$ is the material density, $c = \sqrt{E/\rho}$ is the longitudinal wave speed, and the product $\rho c$ is the material’s characteristic acoustic impedance. This relationship is not an empirical curve fit — it falls directly out of the equations of motion and momentum conservation for a one-dimensional elastic wave, making it as fundamental in its domain as Hooke’s law is in static elasticity.
The practical value of the relationship is that velocity is often the most reliably measured or simulated quantity in a shock or vibration environment — accelerometer data integrates cleanly to velocity, and velocity is comparatively insensitive to high-frequency noise relative to acceleration — while stress at a specific structural location is usually much harder to measure directly and requires either strain gauging at the location of interest or a detailed finite element model.
Strain Energy Density, Revisited
As discussed in a previous post on strain energy density methods in fatigue and fracture, the strain energy density for a uniaxial elastic stress state is:
$$W = \frac{\sigma^2}{2E}$$
This is the same quantity that underlies the Total Strain Energy Density (TSED) fatigue parameter of Ellyin and Kujawski, the Averaged Strain Energy Density (ASED) notch fatigue method of Lazzarin and Berto, and Sih’s strain energy density factor for mixed-mode fracture. In every case, the elastic component of the energy parameter reduces to a simple function of stress squared divided by the elastic modulus.
Combining the Two: A Velocity-Based SED Parameter
Substituting the stress-velocity relationship into the elastic strain energy density expression gives a direct mapping from particle velocity to strain energy density:
$$W = \frac{(\rho c V)^2}{2E} = \frac{\rho^2 c^2 V^2}{2E}$$
Since $c^2 = E/\rho$ for a uniaxial elastic wave, this simplifies further:
$$W = \frac{\rho E V^2}{2E} = \frac{\rho V^2}{2}$$
This is a compact and physically transparent result, and a satisfying one: the elastic strain energy density associated with a one-dimensional stress wave is simply the kinetic energy density of the moving material, $\rho V^2/2$. This is exactly what energy equipartition in an elastic wave would predict — the energy carried by the wave splits evenly between strain (potential) energy and kinetic energy, so the strain energy density alone equals half the total wave energy density, which in this uniaxial form reduces neatly to $\rho V^2/2$. No intermediate stress calculation, elastic modulus, or wave speed is required — only the material density and the measured or predicted particle velocity.
For cyclic or shock loading characterized by a velocity range $\Delta V$ rather than a single peak value, the corresponding energy range follows the same form used in the TSED fatigue framework:
$$\Delta W = \frac{\rho (\Delta V)^2}{2}$$
This $\Delta W$ can be substituted directly into the energy-life relationships already established for metals:
$$\Delta W \cdot N_f^\beta = C_p \quad \text{(Halford-type energy-life form)}$$
or, for the Ellyin-Kujawski total SED formulation that accounts for mean stress effects through the positive elastic energy term:
$$\Delta W_{\text{total}} = \Delta W_p + \Delta W^+$$
where the velocity-derived $\Delta W$ above stands in directly for $\Delta W^+$ when the loading is predominantly elastic, as is typical for high-cycle fatigue under shock and vibration environments.
Why This Matters for Shock and Vibration Engineering
This velocity-based SED formulation offers several practical advantages for structural assessment work where the primary measured or simulated quantity is velocity rather than stress.
Direct use of SRS and velocity-domain data. The shock response spectrum is already expressed in pseudo-velocity terms in many presentations. A velocity-based SED parameter allows fatigue or fracture screening directly from SRS data without first converting to an equivalent static stress through a separate structural model, which is often the most uncertain and time-consuming step in a shock qualification assessment.
Compatibility with machinery vibration severity standards. ISO 10816 and related machinery vibration standards already use velocity as the primary severity metric because velocity correlates well with fatigue damage potential across a broad range of machine types and sizes — this is, in effect, an empirical validation of the same physical principle underlying the velocity-SED formulation derived above.
Mesh-insensitive notch and weld assessment. When this velocity-based energy parameter is combined with the Lazzarin-Berto averaged SED approach — evaluating $\bar{W}$ over a control volume $R_0$ rather than at a singular point — the same mesh-insensitivity advantages documented for the ASED method apply. A coarse dynamic FE model, sufficient to capture global velocity response accurately, can be paired with a control-volume energy evaluation to produce a fatigue life estimate without the fine mesh normally required to resolve notch-tip stress singularities under dynamic loading.
A natural bridging tool for additive manufacturing components under shock loading. As described previously, the ASED method has become a leading tool for fatigue assessment of additively manufactured parts, where defect morphology is too complex to model explicitly. Many AM components — brackets, housings, lattice structures — are increasingly used in mass-optimized aerospace and defense hardware subject to shock and random vibration qualification. A velocity-based SED parameter offers a natural way to connect shock test or simulation output directly to AM-specific fatigue assessment without an intermediate stress-domain translation step.
Worked Example
Consider a 6061-T6 aluminum bracket subjected to a shock event with a peak SRS pseudo-velocity of $V = 5\ \text{m/s}$ at the frequency of interest. Material properties: $\rho = 2700\ \text{kg/m}^3$, $E = 68.9\ \text{GPa}$.
The elastic strain energy density associated with this velocity is:
$$W = \frac{\rho V^2}{2} = \frac{(2700)(5)^2}{2} \approx 3.38\times10^{4}\ \text{J/m}^3$$
For comparison, the equivalent dynamic stress from the stress-velocity relationship, using $c = \sqrt{E/\rho} \approx 5052\ \text{m/s}$:
$$\sigma = \rho c V = (2700)(5052)(5) \approx 68.2\ \text{MPa}$$
Checking consistency: $W = \sigma^2/2E = (68.2\times10^6)^2/(2 \times 68.9\times10^9) \approx 3.37\times10^4\ \text{J/m}^3$, which matches the direct velocity-based calculation above, confirming the formula. This consistency check is good practice any time the velocity-based shortcut is used on a new problem, since an error in units (for example, mixing mm/s and m/s, a common pitfall when working from SRS plots) will show up immediately as a large mismatch between the two calculation paths.
Limitations
This velocity-based SED approach inherits the same limitations as the underlying stress-velocity relationship and the ASED method individually. The stress-velocity relationship strictly applies to one-dimensional elastic wave propagation; for complex three-dimensional structures, it provides a useful bounding estimate rather than an exact local stress, and the same caveat carries through to the derived energy parameter. A related practical caution is that SRS pseudo-velocity is a single-degree-of-freedom oscillator response, not a direct particle velocity measurement; using SRS pseudo-velocity directly in the $\rho V^2/2$ formula without first confirming it reasonably approximates local particle velocity at the structural location and frequency of interest can introduce significant error.
As with the ASED method generally, large-scale plasticity requires correction through Neuber’s rule or the Equivalent Strain Energy Density method, and non-proportional multiaxial loading can reduce the accuracy of simple scalar energy summation. The approach is best suited as a rapid screening and design-margin tool, with detailed FE-based ASED or traditional stress-based fatigue analysis reserved for final qualification of flight or safety-critical hardware.
Summary
The stress-velocity relationship and strain energy density methods are not separate tools but two expressions of the same underlying physics — both reduce a complex structural response to a single scalar quantity derived from elastic wave or energy principles. Substituting $\sigma = \rho c V$ into the elastic strain energy density expression yields a direct, physically transparent path from measured or simulated velocity response to a TSED- or ASED-style fatigue and fracture damage parameter, bypassing the intermediate stress calculation that is often the most uncertain step in shock and vibration qualification analysis. This approach is best used as a fast screening and design-margin tool, complementing rather than replacing detailed stress-based fatigue analysis for final qualification.
References
Irvine, T. (Rev Q). Shock and Vibration Stress as a Function of Velocity. VibrationData.com.
Ellyin, F., Kujawski, D. (1984). Plastic strain energy in fatigue failure. Journal of Pressure Vessel Technology, 106(4), 342–347.
Lazzarin, P., Zambardi, R. (2001). A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. International Journal of Fracture, 112(3), 275–298.
Halford, G.R. (1966). The energy required for fatigue. Journal of Materials, 1, 3–18.
ISO 10816 (now superseded by ISO 20816). Mechanical vibration — Evaluation of machine vibration by measurements on non-rotating parts. International Organization for Standardization.
Sih, G.C. (1974). Strain-energy-density factor applied to mixed mode crack problems. International Journal of Fracture, 10, 305–321.
Tom Irvine | VibrationData.com | Structural Dynamics, Shock, Vibration & Acoustics