# Stress Intensity Factor KI and Crack–hard Inclusion Interaction Study of a Single Edge Cracked Plate by Adaptive Remeshing and Photoelasticity

## Main Article Content

## Abstract

An adaptive remeshing and photoelasticity techniques are presented to determine the stress intensity factors *K _{I}* and crack-hard inclusion interaction of a single edge cracked plate for two-dimensional fracture mechanics problems. The paper starts from describing two-dimensional linear fracture mechanics theory and an adaptive remeshing

*H*method using the quadrilateral and triangular elements. The computational procedure and related finite element equations are explained. The photoelastic theory and its experimental procedure with the use of the stress optic laws are then described. The photoelasticity prototype is designed and built. Performance of adaptive remeshing method is evaluated by analyzing a single edge cracked plate made from polycarbonate. A crack plate with a hard inclusion is then studied for stress intensity factor and crack-hard inclusion interaction. The hard inclusion is made from aluminum. The

*K*stress intensity factor is found to be a function of the crack length per width. The results of adaptive remeshning method and the photoelasticity technique are compared with Brown’s study. This example demonstrates the efficiency of the adaptive remeshing method to provide accurate solutions as compared to those from the photoelastic technique. Then, crack-hard inclusion interaction is studied by varying stress intensity ratio and

_{I}*E*ratio. The crack-hard inclusion interaction behavior is formulated in exponential equation, i.e. cracking tip shielding function. The cracking tip shielding function shows that maximum stress intensity ratio reduces rapidly if

*E*ratio increases. The normalized stress intensity factor is as a convergence exponential function of

*E*ratio.

## Article Details

*Applied Science and Engineering Progress*,

*13*(4), 336–345. Retrieved from https://ph02.tci-thaijo.org/index.php/ijast/article/view/242290

## References

[2] C. Atkinsons, “The interaction between a crack and an inclusion,” International Journal of Engineering Science, vol. 10, no. 2, pp. 127–136, 1972.

[3] B. J. O. Toole and M. H. Santare, “Photoelastic investigation of crack-inclusion interaction,” Experimental Mechanics, vol. 30, no. 3, pp. 253–257, 1990.

[4] S. A. Sushschenko, “Photoelastic analysis of stress concentrations in a two-dimensional model of hard inclusions in a metal matrix,” Tribology Transactions, vol. 40, no. 2, pp. 386–390, 1997.

[5] E. E. Nugent, R. B. Calhoun, and A. Mortensen, “Experimental investigation of stress and strain fields in a ductile matrix surrounding an elastic inclusion,” Acta Materialia, vol. 48, pp. 1451– 1467, 2000.

[6] F. D. Corso, D. Bigoni, and M. Gei, “The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part I. Full–field solution and asymptotics,” Journal of the Mechanics and Physics of Solids, vol. 56, pp. 815–838, 2008.

[7] J. H. Kim and G. H. Paulino, “Simulation of crack propagation in functionally graded materials under mixed–mode and non–proportional loading,” International Journal of Mechanics and Materials in Design, vol. 1, pp. 63–94, 2004.

[8] S. Phongthanapanich and P. Dechaumphai, “Adaptive delaunay triangulation with objectoriented programming for crack propagation analysis,” Finite Element in Analysis and Design, vol. 40, no. 13–14, pp. 1753–1771, 2004.

[9] A. M. Alshoaibi, M. S. A. Hadi, and A. K. Ariffin, “An adaptive finite element procedure for crack propagation analysis,” Journal of Zheijang Univeristy Science A, vol. 8, no. 2, pp. 228–236, 2006.

[10] W. Limtrakarn and P. Dechaumphai, “Adaptive finite element method to determine KI and KII of crack plate with different EINCLUSION/EPLATE ratio,” Transactions of the Canadian Society for Mechanical Engineering, vol. 35, no. 3, pp. 355–368, 2011.

[11] B. N. Rao and S. Rahman, “An efficient meshless method for fracture analysis for cracks,” Computational Mechanics, vol. 26, pp. 398–408, 2000.

[12] L. Luliang and Z. Pan, “Meshless method for 2D mixed–mode crack propagation based on voronoi cell,” ACTA Mechanica Solida Sinca, vol. 16, no 3, pp. 231–239, 2003.

[13] Y. J. Chiuo, Y. M. Lee, and R. J. Tsay, “Mixed mode fracture propagation by manifold method,” International Journal of Fracture, vol. 114, pp. 327–347, 2002.

[14] K. Xu, S. T. Lie, and Z. Cen, “Crack propagation analysis with galerkin boundary element method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 22, no. 5, pp. 421– 435, 2004.

[15] R. Kitey, A. V. Phan, H. V. Tippur, and T. Kaplan, “Modeling of crack growth through particulate clusters in brittle matrix by symmetric–galerkin blundary element method,” International Journal of Fracture, vol. 141, pp. 11–25, 2006.

[16] M. R. Ayatollahi and H. Safari, “Evaluation of crack tip constraint using photoelasticity,” International Journal of Pressure Vessels and Piping, vol. 80, no. 9, pp. 665–670, 2003.

[17] S. W. Nam, Y. W. Chang, S. B. Lee, and N. J. Kim, “New Insights into plasticity-induced crack tip shielding via mathematical modelling and full field photoelasticity,” Key Engineering Materials, vol. 345–346, pp. 199–204, 2007.

[18] P. C. Savalia and H. V. Tippur, “A study of crack-inclusion interactions and matrix-inclusion debonding using moire interferometry and finite elment method,” Experimental Mechanics, vol. 47, no. 4, 2007.

[19] C. J. Christopher, M. N. James, K. F. Tee, and E. A. Patterson, “Evaluation of crack tip shielding using a photoelastic model,” in Proceeding the SEM XI International Congress on Experimental and Applied Mechanics, pp. 371–380, 2008.

[20] C. J. Christophera, M. N. Jamesa, E. A. Patterson, and K. F. Teea, “A quantitative evaluation of fatigue crack shielding forces using photoelasticity,” Engineering Fracture Mechanics, vol. 75, no. 14, pp. 4190–4199, 2008.

[21] M. N. James, Y. W. Lu, C. J. Christopher, and E. A. Patterson, “Full-field modelling of crack tip shielding via the ‘Plastic Inclusion’ concept,” Advanced Materials Research, vol. 118–120, pp. 1–9, 2010.

[22] M. N. James, C. J. Christopher, Y. W. Lu, K. F. Tee, and E. A. Patterson, “Crack tip shielding from a ‘Plastic Inclusion’,” Key Engineering Materials, vol. 465, pp. 1–8, 2011.

[23] S. Bhat and S. Narayanan, “A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces,” Acta Mechanica, vol. 216, no. 1–4, pp. 259–279, 2011.

[24] T. Hiroshi, C. P. Paul, and R. I. George, The Stress Analysis of Crack Handbook. New York: ASME, 1985, pp. 46–54.

[25] T. L. Anderson, Fracture Mechanics: Fundamentals and Applications. Florida: CRC Press, 1994.

[26] W. Limtrakarn, “Stress analysis on crack tip using Q8 and adaptive meshes,” Thammasat International Journal of Science and Technology, vol. 10, no. 1, pp. 19–24, 2005.

[27] O. C. Zienkiewicz and R. L. Taylor, Finite Element Method, 5th ed., Oxford, England: Butterworth– Heinemann, 2000.

[28] G. V. Guinea, J. Planas, and M. Elices “KI evaluation by the displacement extrapolation technique,” Engineering Fracture Mechanics, vol. 66, no. 3, pp. 243–255, 2000.

[29] R. Ramakrishnan, K. S. Bey, and E. A. Thornton, “Adaptive quadrilateral and triangular finiteelement scheme for compressible flows,” AIAA Journal, vol. 28, no. 1, pp. 51–59, 1990.

[30] P. Dechaumphai and E. A. Thornton, “Improved Finite Element Methodology for Integrated Thermal-Structure Analysis,” NASA, Washington, DC, NASA CR 3635, Nov.1982.

[31] R. Lohner, K. Morgan, and O. C. Zienkiewicz, “Adaptive grid refinement for the Euler and compressible Navier-Stokes equations,” in Proceedings the International Conference on Accuracy Estimates and Adaptive Refinements in Finite Element Computations, 1984, vol. 2, pp. 189–202.

[32] P. Dechaumphai, “Adaptive finite element technique for heat transfer problems,” Journal of Energy Heat and Mass Transfer, vol. 17, no. 2, pp. 87–94, 1995.

[33] J. Peraire, M. Vahjdati, K. Morgan, and O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computation,” Journal of Computational Physics, vol. 72, pp. 449–466, 1987.

[34] J. T. Oden and G. F. Carey, Finite Elements: Mathematical Aspects. New Jersey: Prentice- Hall, 1981.

[35] G. R. Irwin, “Discussion of the dynamic stress distribution surrounding a running crack – A photoelastic analysis,” in Proceedings of the Society for Experimental Stress Analysis, 1958, vol. 16, no. 1, pp. 93–96.

[36] S. A. Paipetis and G. S. Holister, Photoelasticity in Engineering Practice. Netherlands: Elsevier Applied Science Publishers, pp. 181–204, 1985.

[37] M. A. Schroedl and C. W. Smith, Local Stress near Deep Surface Flaws under Cylindrical Bending Fields, Pennsylvania: ASTM International, pp. 45–63, 1973.