Efficient Computation of Dynamic Stress Intensity Factors for Problems with Stationary Cracks Subjected to Time Dependent Loading

Abstract

Purpose

In the current paper, a modal analysis based model order reduction method (MOR) is applied to reduce the computational effort associated with estimating the dynamic stress intensity factors (DSIFs) in three dimensional fracture problems. In addition, a method to select appropriate modes for model reduction is also proposed.

Method

The proposed method is based on the mode acceleration method (MAM), commonly used in structural dynamics. Appropriate mode shapes for carrying out the reduction are chosen based on the magnitude of the modal stress intensity factors associated with them. The modal stress intensity factors are calculated using the displacement field associated with the stiffness normalized modes in the vicinity of the crack front. The dynamic stress intensity factors are then calculated by expressing them as a linear combination of the modal stress intensity factors.

Results

The proposed method is used to solve a representative three-dimensional fracture mechanics problem with stationary crack subjected to various dynamic loading conditions. The results indicate that the proposed method is approximately one order magnitude more accurate than the method described in literature. The results also indicate that the proposed mode selection criteria is effective in identifying contributing modes to the dynamic stress intensity factors in case of low frequency loading.

Conclusions

The results indicate that the proposed method can be used to estimate the dynamic stress intensity factors for low frequency forcing loading quite efficiently. 

Appendix

Solution of a single degree of spring mass system subjected to ramp load followed by a sinusoidal load

Consider a single degree of freedom

$$\begin{aligned} \ddot{u}+\omega _{n}^{2}u=F\left( t\right) \end{aligned}$$

with $u\left( 0 \right) = 0$ and $\dot{u}\left( 0 \right) =0$. If the loading $F\left( t \right)$ is of the form (see Fig. 5)

$$\begin{aligned} F(t)&= \frac{F_0 t}{t_0}, \ \ 0 \le t \le t_0 \\&=F_0 +Asin(\omega (t-t_0)), \ \ t > t_0 \end{aligned}$$

then the solution $u\left( t \right)$ is given by:

$$\begin{aligned} u(t)&= \frac{F_0 t_0}{t_0 \omega _n^2} -\frac{F_0sin(\omega _nt)}{t_0\omega _n^3}, \ \ 0 \le \ \ t \le t_0 \\&=\frac{F_0 }{\omega _n^2} - \frac{F_0sin(\omega _nt)}{t_0\omega _n^3} +\frac{F_0sin(\omega _ n(t-t_0))}{t_0\omega _n^3} \\&\quad - A\frac{\omega _nsin(\omega (t-t_0)) -\omega (sin(\omega _n(t-t_0)))}{\omega _n (\omega ^2-\omega _n^2)}, \ \ t > \ \ t_0 \end{aligned}$$