Stress Analysis of a Perforated Asymmetrical Vehicle Cooling Module Structure from Unidirectional DIC Displacement Information

Abstract

The stresses in a perforated asymmetrical cooling module of a heavy commercial vehicle are determined by processing Digital Image Correlation (DIC)-recorded unidirectional displacement data with a series representation of an Airy stress function. Typical of many real engineering problems, the external loads and boundary conditions applied to the cooling module are not well known, thereby challenging the determination of an accurate analytical or Finite Element Method (FEM) Solution. On the other hand, full-field stress information could potentially be obtained from DIC measured displacements. This would traditionally involve differentiating measured displacement data; a process which can be ill-conditioned and adversely influenced by data noise and quality. A hybrid approach which processes the recorded displacement data using a stress function and determines stresses in finite members has been published previously, but it was restricted to symmetrically loaded structures. The earlier concepts are extended to analyze perforated and asymmetrical isotropic structural member from DIC recorded values of a single component of the displacement field and without having to physically differentiate the measured displacement data.

Appendix

$$ v{=}\frac{1}{E}\left[\fontsize{8.5}{10}\selectfont\begin{array}{@{}l@{}}\left(\left(-\frac{1}{r}\cdot \sin \left(\theta \right)-\frac{3}{4}\frac{r^2}{R^3}\cdot \left( sin\left(\theta \right)-3\cdot \theta \cos \left(\theta \right)\right)\right)\cdot \left(1+v\right)+\left(\frac{r^4}{8\cdot {R}^5}\right)\cdot \left(\left(1+5\cdot v\right)\cdot sin\left(\theta \right)-\left(21+9\cdot v\right)\cdot \theta \cdot cos\left(\theta \right)\right)\right)\cdot {b}_0\hfill \\ {}+\left(2\cdot r\cdot \left(1-v\right)\cdot sin\left(\theta \right)-\frac{3\cdot {r}^2}{2\cdot R}\cdot \left(1+v\right)\cdot \left( \sin \left(\theta \right)-3\cdot \theta cos\left(\theta \right)\right)+\frac{r^4}{4\cdot {R}^3}\cdot \left( sin\left(\theta \right)\cdot \left(1+5\cdot v\right)-\theta \cdot cos\left(\theta \right)\cdot \left(21+9\cdot v\right)\right)\right)\cdot {c}_0\hfill \\ {}+\left(\begin{array}{l}\frac{r^2\cdot \left(1+v\right)}{4\cdot {R}^3}\left( sin\left(\theta \right)\cdot tan\left(3\cdot \theta \right)+ ln\left( cos\left(3\cdot \theta \right)\right)\cdot cos\left(\theta \right)\right)\hfill \\ {}-\frac{r^4}{8\cdot {R}^5}\left( \sin \left(\theta \right)\cdot \tan \left(3\cdot \theta \right)\cdot \left(1+5\cdot v\right)+\frac{\left(21+9\cdot v\right)}{3} \ln \left( cos\left(3\cdot \theta \right)\right)\cdot cos\left(\theta \right)\right)\hfill \end{array}\right)\cdot {A}_0\hfill \\ {}-\left(\frac{R^4\cdot \left(1+v\right)}{r^2}\cdot \cos \left(2\cdot \theta \right)+{r}^2\cdot \left(2\cdot \left(1+v\right)+\left(3-v\right)\cdot cos\left(2\cdot \theta \right)\right)\right)\cdot {d}_1^{\prime }+\left(\frac{R^4\cdot \left(1+v\right)}{r^2}\cdot \sin \left(2\cdot \theta \right)+{r}^2\cdot \left(\left(3-v\right)\cdot sin\left(2\cdot \theta \right)\right)\right)\cdot {d}_1\hfill \\ {}+\left(r\cdot {R}^2\cdot \left(1+v\right)\cdot \left(3\cdot \cos \left(\theta \right)-{r}^{-4}\cdot {R}^4\cdot \cos \left(3\theta \right)\right)-\left(2\cdot \cos \left(\theta \right)\cdot {r}^3\left( cos\left(2\theta \right)\cdot \left(3-v\right)+2v\right)\right)\right){b}_2^{\prime}\hfill \\ {}+\left(\left({R}^{-2}\cdot r\cdot \cos \left(\theta \right)+{R}^2\cdot {r}^{-3}\cdot cos\Big(3\cdot \theta \right)\right)\cdot \left(1+v\right)+2\cdot cos\left(\theta \right)\cdot {r}^{-1}\left(- \cos \left(2\cdot \theta \right)\cdot \left(1+v\right)+2\Big)\right)\cdot {d}_2^{\prime}\hfill \\ {}+\left(\frac{12\cdot {r}^2\cdot \left(1+v\right)}{R^6}\cdot \cos \left(2\cdot \theta \right)+\frac{24\cdot {r}^4}{R^8}\left(-\left(3-v\right)\cdot co{s}^4\left(\theta \right)+2\cdot \left(1-v\right)\cdot co{s}^2\left(\theta \right)+\left(\frac{1+5\cdot v}{8}\right)\right)-\frac{3}{r^4}\left( cos\left(4\cdot \theta \right)\cdot \left(1+v\right)\right)\right)\cdot {c}_3^{\prime}\hfill \\ {}+\left(\begin{array}{l}\frac{9\cdot {r}^2\cdot \left(1+v\right)}{R^4}\cdot \cos \left(2\cdot \theta \right)-\frac{2\cdot {r}^4}{R^6}\left(8\cdot \left(3-v\right)\cdot co{s}^4\left(\theta \right)-16\cdot \left(1-v\right)\cdot co{s}^2\left(\theta \right)-\left(1+5\cdot v\right)\right)\hfill \\ {}+\frac{1}{r^2}\left(-16\cdot \left(1+v\right)\cdot co{s}^4\left(\theta \right)+2\cdot co{s}^2\left(\theta \right)\cdot \left(11+7\cdot v\right)-\left(5+v\right)\right)\hfill \end{array}\right)\cdot {d}_3^{\prime}\hfill \\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot cos\left(\left(n-1\right)\cdot \theta \right)\hfill \\ {}-{r}^{n+1}\cdot \left(\left(n\cdot \left(1+v\right)\right)\cdot cos\left(\left(n-1\right)\cdot \theta \right)-2\cdot sin\left(n\cdot \theta \right)\cdot sin\left(\theta \right)\cdot \left(1-v\right)+4\cdot cos\left(n\cdot \theta \right)\cdot cos\left(\theta \right)\right)\hfill \\ {}-{r}^{-\left(n+1\right)}\cdot {R}^{\left(2n+2\right)}\cdot \left(1+v\right)\cdot cos\left(\left(n+1\right)\cdot \theta \right)\hfill \end{array}\right){b}_n^{\prime}\right]}\hfill \\ {}+{\displaystyle \sum_{n=4,5,6}^N\left[\left(\begin{array}{l}{r}^{\left(n-1\right)}\cdot {R}^{-\left(2n-2\right)}\cdot \left(1+v\right)\cdot cos\left(\left(n-1\right)\cdot \theta \right)\hfill \\ {}-\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot cos\left(\left(n+1\right)\cdot \theta \right)\hfill \\ {}-{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v\right)\cdot cos\left(n+1\right)\cdot \theta \right)-2\cdot \left(1-v\right)\cdot sin\left(n\cdot \theta \right)\cdot sin\left(\theta \right)-4\cdot cos\left(n\cdot \theta \right)\cdot cos\left(\theta \right)\Big)\hfill \end{array}\right)\cdot {d}_n^{\prime}\right]}\hfill \\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-\left(n+1\right)\cdot {r}^{\left(n-1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot sin\left(\left(n-1\right)\cdot \theta \right)\hfill \\ {}+{r}^{n+1}\cdot \left(\left(n\cdot \left(1+v\right)\left)\cdot sin\left(\left(n-1\right)\cdot \theta \right)+2\cdot cos\left(n\cdot \theta \right)\cdot sin\left(\theta \right)\cdot \left(1-v\right)+4\cdot sin\left(n\cdot \theta \right)\cdot cos\right(\theta \right)\right)\hfill \\ {}+{r}^{-\left(n+1\right)}\cdot {R}^{\left(2n-2\right)}\cdot \left(1+v\right)\cdot sin\left(\left(n+1\right)\cdot \theta \right)\hfill \end{array}\right){b}_n\right]}\hfill \\ {}+{\displaystyle \sum_{n=2,3,4}^N\left[\left(\begin{array}{l}-{r}^{\left(n-1\right)}\cdot {R}^{-\left(2n-2\right)}\cdot \left(1+v\right)\cdot sin\left(\left(n-1\right)\cdot \theta \right)\hfill \\ {}+\left(1-n\right)\cdot {r}^{-\left(n+1\right)}\cdot {R}^2\cdot \left(1+v\right)\cdot sin\left(\left(n+1\right)\cdot \theta \right)\hfill \\ {}+{r}^{\left(-n+1\right)}\left(\left(n+n\cdot v\right)\cdot sin\left(\left(n+1\right)\cdot \theta \right)+2\cdot \left(1-v\right)\cdot cos\left(n\cdot \theta \right)\cdot sin\left(\theta \right)-4\cdot sin\left(n\cdot \theta \right)\cdot cos\left(\theta \right)\right)\hfill \end{array}\right)\cdot {d}_n\right]}\hfill \end{array}\right] $$