Damage Rate Laws and Failure Statistics for Lumped Coupled-Field Systems via Averaging

TL;DR

The paper develops averaged damage evolution rate laws for a coupled-field fatigue model to efficiently study long-time failure statistics. By separating fast macroscopic dynamics from slow damage evolution through the method of averaging, it derives tractable rate laws for 1- and 2-DOF lumped systems, enabling both analytical and Monte Carlo analyses of failure times and modes. The work identifies brittle-like regimes (certain $p,q$ exponents) where damage growth is delayed and then abrupt, increasing time-variance while reducing location-variance, and constructs phase-space portraits with separatrices to predict failure mode probabilities. Overall, the approach delivers significant computational speed-ups and broad analytical insights with potential applicability to more complex multi-DOF damage models.

Abstract

We study the non-linear dynamics and failure statistics of a coupled-field fatigue damage evolution model. We develop a methodology to derive averaged damage evolution rate laws from such models. We show that such rate laws reduce life-cycle simulation times by orders of magnitude and permit dynamical systems analysis of long-time behavior, including failure time statistics. We use the averaged damage rate laws to study 1 DOF and 2 DOF damage evolution models. We identify parameter regimes in which the systems behave like a brittle material and show that the relative variability for failure times is high for such cases. We also use the averaged rate laws to construct damage evolution phase portraits for the 2 DOF system and use insights derived from them to understand failure time and location statistics. We show that, for brittle materials, as the relative variability in failure time increases, the variability in failure location decreases.

Figures (11)

• Figure 1: (a) A spring-mass-damper model of a general body. (b) A 1 DOF spring-mass-damper model under periodic external load. In either case the stiffness of the springs alters as damage evolves
• Figure 2: (left-hand side plot) Comparison of damage evolution curves obtained by integrating the averaged and the fully-coupled system. (curve on left) Parameter values: $\omega = 1$, $\omega_n = 0.5$, $\eta = 0.0001$, $\zeta = 0.003$, $F = 2.5$ and $\phi_0 = 1 \times 10^{-6}$. Time-step size for either simulation : $\Delta t = \Delta \tau = 5\times 10^{-4}$. (curve on right) A different damage evolution curve that is obtained when there is an early passage through resonance. Parameter values: $\omega = 1$, $\omega_n = 1.1$, $\eta = 0.0001$, $\zeta = 0.1$, $F = 3.0$ and $\phi_0 = 1\times 10^{-6}$. (right-hand side plot) Damage evolution curves for different values of $p$ and $q$. Parameter values: $\omega = 1$, $\omega_n = 0.5$, $\eta = 0.0001$, $\zeta = 0.003$, $F = 2.5$ and $\phi_0 = 1 \times 10^{-6}$
• Figure 3: (left)Comparison of failure time distributions obtained using the averaged and fully-coupled system. Parameter values: $F =6$, $\omega = 1$, $\omega_n = 0.5$, $\eta = 0.0001$, $p=1$, $q=-1$ and $\zeta = 0.003$. $\phi_0$ is uniformly distributed on $[0,\;1\times 10^{-6}]$(right) CDF for same parameters plotted on an exponential plot.
• Figure 4: Exponential plot of failure time CDFs for different values of $p$. Other parameter values same as in Fig. \ref{['CDF']}
• Figure 5: (left) Pseudocolor plot showing the variation of relative temporal variability of the failure time with $p$ and $\omega$.(right)$F$-$N$ curve. Parameter values: $\omega_n=0.5,\, \zeta=3\times 10^{-3},\, \hat{q}=1,\, \phi_0\sim$ Unif $[0\,,1\times 10^{-6}]$, No.of samples = $1\times 10^6$, $\omega=1$. For the right plot, $\alpha = 5\times10^{-3}$, other parameters are the same as in the left plot.