Finite Elements Method in Fracture Mechanics Naoto Sakakibara

outline Introduction Collapsed Quadrilateral QPE Element Enriched Element Demo – NS-FFEM 1. 0 Result Extended Finite Element Method Summary

FEM in Fracture Mechanics Early Application for Fracture Mechanics > 5 -10% error for simple problem *1 > solutions around tip cannot guaranteed*2 a. Crack tip element – Quarter Point Element b. Enriched Element – Add another DOF

Collapsed Quarter Point Element • Henshell and Shaw, 1975 • 1/√r variation for strain can be achieved • Same shape function N, • Standard FEM can be used • Collapsed Element, more accuracy than other QPEs. Ex) 7 4 3 473 H/4 8 6 3 H/4 1 5 2 0 0. 2 0. 4 0. 6 0. 8 1

Transition Element Lynn and Ingraffea, 1978 Combined with QPE element Improving the accuracy of SIF, under special configuration Located between Normal Element & QPE Collapsed QPE (1, 0) (βL, 0) (L, 0)

Meshing tips Suggestion • L-QPE/ 4 a ~ 0. 05 -0. 2 • L-QPE/L-Tra. ~ 1. 5244 • Number of QPE ~ 6 – 12 a L -Tra. L -QPE Quarter Point Element Transitional Element Isoparametric Element Note: No optimal element size!

Enrich Element • Adding the analytic expression of the crack tip field to the conventional FEM General FEM Drawbacks • Additional DOF Not able to use general FEM • Higher order more integration point • Incompatibility in displacement Transition element Singular field term Part of the solution of displacement field

NS-FFEM ver 1. 0 Method • Gaussian Elimination • Algebraic BC B, D Input • CPE 4, CPE 8, QPE 8+Transitional • Mesh number • Geometry • Material Property Output • SIF (QPDT) • σ, ε • u, v

Deformed Configuration ABAQUS QPE with CPE 8 NS-FFEM with QPE

Result-1 SIF QPDT method SIF DCT method E C D B

Result - 2 Enriched by singular function around tip.

Extended FEM-1 F - Singular field function H – Discontinuous function EI EII D A II I B FI FII C • H – step, sign, etc. • εI(x), εII(x) – different function • a – associated with displacements at E & F • Mesh – independent from crack

Extended FEM-2 Discontinuous Function H Singular field Function

Summary 1. QPE Transitional • DOF ~ # of nodes • Mesh size, no optimal size • Mesh, depend on crack 2. Singular Field • Additional DOF • More Integration point at crack tip element • Mesh, depend on crack 3. Singular Field Discontinuous • Additional DOF • Mesh, independent from crack • No remeshing for crack growth

Reference Chona, R. , Irein, G. , and Sanford, R. J. (1983). The influence of specimen size and shape on the singurarity-dominated zone. Proceedings, 14 th National Symposium on Fracture Mechanics, STP 791, Vol. 1, American Soc. for Testing and Materials, (pp. I 1 -I 23). Philadelphia. I. L. Lim, I. W. Jhonston and S. K. Choi. (1993). Application of singular quadratic distorted isoparametric elements in linear fracture mechanics. International journal for numerical methods in engineering , Vol. 36, 2473 -2499. I. L. Lim, I. W. Johnston and S. K. Choi. (1992). On stress intensity factor computation from the quater-point element displacements. Communications in applied numerical methods , Vol. 8, 291 -300. Mohammad, S. (2008). Extendet finite element. Blackwell Publishing. Nicolas Moes, John Dolbow and Ted Belystschko. (1999). A finite element method for crack growth withiout remeshing. International jounarl for numerical methods in engineering , 131 -150. Sanford, R. (2002). Principle of Fracture Mechanics. Upper Saddle River, NJ 07458: Pearson Education, Inc.