### A Cell Centered Finite Volume Formulation for the Calculation of Stress Intensity Factors in Mindlin-Reissner Cracked Plates

Abuzar Amraei, Nosratollah Fallah

#### Abstract

In fracture analysis, the stress intensity factor (SIF) is an important parameter which is needed for describing the stress state at crack tip. In this paper a finite volume formulation is developed for calculating the stress intensity factor (SIF) in Mindlin-Reissner plates with a through-the-thickness crack (through crack). For approximating the field variables and its derivatives the moving least square (MLS) technique is utilized. The problem domain is discretized into a mesh of elements where each element is considered as a control volume (CV). The center of CVs are considered as computational points where the unknown variables are associated with. The equilibrium equations of each CV are written based on the stress resultant forces acting on the boundaries of CV where the first order shear deformation theory (FSDT) is implemented in the formulation. Some benchmark problems of plate with through cracks are solved by the present method and the obtained results are compared with the results of analytical and XFEM numerical methods in order to demonstrate the accuracy of the present formulation. These comparisons illustrate the accuracy of predictions of the present solution method. Nevertheless, it is found that the formulation is free of shear locking property which greatly facilitates the cracked plates analysis due to its dual capabilities of analyzing both thin and moderately thick cracked plates.

#### Keywords

Mindlin-Reissner Plate Theory; Finite Volume Method; Moving Least Squares.

#### References

Williams M L. “The bending stress distribution at the base of a stationary crack.” Journal of Applied Mechanics 28 (1961): 78–82. https://doi:10.1115/1.3640470.

Keer K, and Sve C. “On the bending of cracked plates.” Int. J. Solids Struct 6(1970): 1545-1559. https://doi:org/10.1016/0020-7683(70)90063-6.

Knowles J. and Wang N. “On the bending of an elastic plate containing a crack“ Journal of Mathematics and Physics 39(1960): 223-236. https:// doi: 10.1002/sapm1960391223.

Hartranft RJ. and Sih GC. “Effect of plate thickness on the bending stress distribution around through cracks“ Journal of Mathematics and Physics 47(1968): 276-291. https:// doi:10.1002/sapm1968471276.

Boduroglu H, and Erdogan F.” Internal and edge cracks in a plate of finite width under bending” ASME J. Appl.Mech, 50 (1983): 621-629. https://doi:10.1115/1.3167100.

Joseph P, Erdogan F.” Bending of a thin Reissner plate with a through crack” Journal of Applied Mechanics 58 (1991): 842-846. https:// doi:10.1115/1.2897273.

Zucchini A, Hui CY, and Zehnder AT.” Crack tip stress fields for thin plates in bending, shear and twisting: A comparison of plate theory and three dimensional elasticity theory” International Journal of Fracture 104 (2000): 387-407. https:// doi.org/10.1023/A:1007699314793.

Hui CY and Zehnder A.” A theory for the fracture of thin plates subjected to bending and twisting moments” International Journal of Fracture 61 (1993): 211-229. https://doi.org/10.1007/BF00036341.

Zehnder A. T. and Viz M. J., ”Fracture Mechanics of Thin Plates and Shells Under Combined Membrane, Bending and Twisting Loads” Applied Mechanics Reviews 58 (2005): 37-48. https:// doi:10.1115/1.1828049.

Wang Y.H., Tham L.G., Lee P.K.K., and Tsui Y. ”A boundary collocation method for cracked plates” Comput. Struct. 81(2003): 2621–2630. https:// doi.org/10.1016/S0045-7949(03)00324-9.

Alwar R. S., and Nambissan K. N. R.” Three-dimensional finite element analysis of cracked thick plates in bending” Int. J. Num. Meth. Engng 19 (1983): 293-303. https:// doi:10.1002/nme.1620190210.

Barsoum RS.” A degenerate solid element for linear fracture analysis of plate bending and general shells” International Journal for Numerical Methods in Engineering 10 (1976): 551-564. https:// doi:10.1002/nme.1620100306.

Ahmad J. and Loo FTC. ”Solution of plate bending problems in fracture mechanics using a specialized finite element technique” Engineering Fracture Mechanics 11(1979): 661-673. https:// doi.org/10.1016/0013-7944 (79)90127-9.

Alwar RS. and Ramachandran KNN. ”Three-dimensional finite element analysis of cracked thick plates in bending” International Journal for Numerical Methods in Engineering 19(1983): 293-303. https://doi:10.1002/nme.1620190210.

Rhee1 H. C, and Atluri S. N. “Hybrid stress finite element analysis of bending of a plate with a through flaw.” International Journal for Numerical Methods in Engineering 18 (1982): 259–271. https:// doi:10.1002/nme.1620180208.

Sosa H. A. and Eischens J W. “Computation of stress intensity factors for plate bending via a path- independent integral” Engineering Fracture Mechanics 25 (1986): 451-462. https:// doi.org/10.1016/0013-7944(86)90259-6.

Dolbow J, Moes N, and Belytschko T. “Modeling fracture in Mindlin-Reissner plates with the extended finite element method” International Journal of Solids and Structures 37(2000):7161-7183.https://doi.org/10.1016/S0020-7683(00)00194-3.

Su R.K.L., Leung A.Y.T. “Mixed mode cracks in Reissner plates“ Int. J. Fract. 107 (2001):235–257. https:// doi.org/10.1023/A:1007652028645.

Lasry J, Renard Y, and Salaun M. ”Stress Intensity Factors computation for bending plates with XFEM” International Journal for Numerical Methods in Engineering 00 (2010): 1–21. https:// doi: 10.1002/nme.

Bhardwaj G, Singh I. V., Mishra B. K., Virender Kumar. “Numerical Simulations of Cracked Plate using XIGA under Different Loads and Boundary Conditions” Mechanics of Advanced Materials and Structures 23 (2016):704-714. https:// doi.org/10.1080/15376494.2015.1029159.

Tanaka S., Suzuki H., Sadamoto S., Imachi M., and Bui Q. T., “ Analysis of cracked shear deformable plates by an effective meshfree plate formulation“ Eng. Fract. Mech. 144 (2015)142–157. https:// doi.org/10.1016/j.engfracmech.2015.06.084.

. Tanaka S., Suzuki S., Sadamoto S., Okazawa Yu T. T., and Bui T. Q. “Accurate evaluation of mixed-mode intensity factors of cracked shear-deformable plates by an enriched meshfree Galerkin formulation“ Archive of Applied Mechanics 87 (2017): 279–298. https://doi.org/10.1007/s00419-016-1193-x.

Onate E, Cervera M, and Zienkiewicz O. C. “A finite volume format for structural mechanics” Int Centre for Numerical Methods in Engineering, Int. J. Numer. Methods Eng. 15(1992). https:// doi:10.1002/nme.1620370202.

Wheel M.A. “A ﬁnite volume method for analyzing the bending deformation of thick and thin plates” Comput. Methods Appl. Mech. Eng. 147 (1997): 199–208. https:// doi.org/10.1016/S0045-7825(96)00003-5.

Fallah N. “A cell vertex and cell centred ﬁnite volume method for plate bending analysis” Comput. Methods Appl. Mech. Engrg. 193 (2004) 3457–3470. https:// doi.org/10.1016/j.cma.2003.08.005.

Ivankovic A, Demirdzic I, Williams, J.G, and Leevers P.S. “Application of the ﬁnite volume method to the analysis of dynamic fracture problems” International Journal of Fracture 66 (1994): 357. https:// doi.org/10.1007/BF00018439.

Stylianou V.P, and Ivankovic A. “Finite volume analysis of dynamic fracture phenomena. Part I: A node release methodology” International Journal of Fracture 113 (2002): 107–123. https:// doi.org/10.1023/A:1015532129150 .

Stylianou V.P, and Ivankovic A. “Finite volume analysis of dynamic fracture phenomena. Part II. A cohesive zone type methodology” International Journal of Fracture 113 (2002): 125–151. https:// doi.org/10.1023/A:1015563602317 .

Atluri SN, and Zhu T. “A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics” Computational Mechanics 22 (1998a): 117-127. https:// doi.org/10.1007/s004660050346.

Belytschko T, Lu YY, and Gu L.“Element-free Galerkin methods.” Int.J.Numer. Methods Eng 37 (1994a): 229-256. https://doi: 10.1002/nme.1620370205.

Liu G.R. “Meshfree methods: moving beyond the finite element method“ CRC Press (2010). https://doi: 10.1002/nme.1620370205.

Atluri S. N & Shen Sh. “The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less-costly Alternative to the Finite Element and Boundary Element Methods” CMES 3 (2002):11-51. https:// doi:10.3970/cmes.2002.003.011.

Moosavi MR, Khelil A. “Accuracy and computational efﬁciency of the ﬁnite Volume method combined with the meshless local Petrov–Galerkin in Comparison with the ﬁnite element method in elasto-static problem.” ICCES 5(2008);211–38.

Moosavi M. R, F. Delfanian, Khelil A, Khelil A. “Orthogonal meshless ﬁnite volume method applied to crack problems” Thin-Walled Structures, 52 (2012): 61-65. https:/ /doi.org/10.1016/j.tws.2011.10.009.

Ebrahimnejad M, Fallah N, Khoei A. R. “New approximation functions in the meshless finite volume method for 2D elasticity problems” Engineering Analysis with Boundary Elements 46 (2014):10–22. https:// doi.org/10.1016/j.enganabound.2014.04.023.

Qian L. F, Batra R. C, and Chen L. M. “Elastostatic Deformations of a Thick Plate by using a Higher-Order Shear and Normal Deformable Plate Theory and two Meshless Local Petrov-Galerkin (MLPG) Methods” CMES 4(2003):161-175.

Belinha J, Dinis L.M.J.S. “Analysis of plates and laminates using the element-free Galerkin method” Computers and Structures 84 (2006):1547–1559. https:// doi.org/10.1016/j.compstruc.2006.01.013.

Dinis L.M.J.S, Natal Jorge R.M, Belinha J. “Analysis of plates and laminates using the natural neighbour radial point interpolation method” Engineering Analysis with Boundary Elements 32 (2008):267–279. https:// doi.org/10.1016/j.enganabound.2007.08.006.

Mindlin R D. “Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates” ASME J. Appl. Mech (1951) 73: 31.

Fleming M, Chu YA, Moran B. Belytschko T, ”Enriched element-free Galerkin methods for crack tip fields” Int. J. Numer. Methods Eng. 40(1997):1483–1504. https://doi:10.1002/(SICI)1097-0207(19970430)40:8<1483::AID-NME123>3.0.CO;2-6.

Hartranft R. J, and Sih G. C. “The use of eigenfunction expansions in the general solution of the three-dimensional crack problems” J. Math. Mech 19 (1969): 123-138. https:// www.jstor.org/stable/24901923.

Wilson W. K, and Thompson D. G. “On the finite element method for calculating stress intensity factors for cracked plates in bending” Engineering Fracture Mechanics 3 (1971): 97-102. https:// doi.org/10.1016/0013-7944(71)90001-4.

Bayesteh H, and Mohammadi S. “XFEM fracture analysis of shells: The effect of crack tip enrichments, Computational Materials Science 50 (2011): 2793–2813. https:// doi.org/10.1016/j.commatsci.2011.04.034

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DOI: 10.28991/cej-030951

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