A physically and geometrically nonlinear scaled‐boundary‐based finite element formulation for fracture in elastomers

SUMMARY This paper is devoted to the formulation of a plane scaled boundary finite element with initially constant thickness for physically and geometrically nonlinear material behavior. Special two-dimensional element shape functions are derived by using the analytical displacement solution of the standard scaled boundary finite element method, which is originally based on linear material behavior and small strains. These 2D shape functions can be constructed for an arbitrary number of element nodes and allow to capture singularities (e.g., at a plane crack tip) analytically, without extensive mesh refinement. Mapping these proposed 2D shape functions to the 3D case, a formulation that is compatible with standard finite elements is obtained. The resulting physically and geometrically nonlinear scaled boundary finite element formulation is implemented into the framework of the finite element method for bounded plane domains with and without geometrical singularities. The numerical realization is shown in detail. To represent the physically and geometrically nonlinear material and structural behavior of elastomer specimens, the extended tube model and the Yeoh model are used. Numerical studies on the convergence behavior and comparisons with standard Q1P0 finite elements demonstrate the correct implementation and the advantages of the developed scaled boundary finite element. Copyright © 2014 John Wiley & Sons, Ltd.

[1]  Francis Tin-Loi,et al.  A scaled boundary polygon formulation for elasto-plastic analyses , 2014 .

[2]  Bin Teng,et al.  A modified scaled boundary finite-element method for problems with parallel side-faces. Part I. Theoretical developments , 2005 .

[3]  Zhao Xing-zhong,et al.  Ferroelectric properties of Bi 25 La 0.75 Ti 3 O 12 thin films prepared by sol-gel method , 2007 .

[4]  Ean Tat Ooi,et al.  A hybrid finite element-scaled boundary finite element method for crack propagation modelling , 2010 .

[5]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[6]  Zora Vrcelj,et al.  Evaluation of dynamic stress intensity factors and T-stress using the scaled boundary finite-element method , 2008 .

[7]  Zhenjun Yang,et al.  RECENT PROGRESS IN MODELING CRACK PROPAGATION USING THE SCALED BOUNDARY FINITE ELEMENT METHOD , 2012 .

[8]  J. Wolf,et al.  Consistent infinitesimal finite-element―cell method: out-of-plane motion , 1995 .

[9]  J. Wolf,et al.  A virtual work derivation of the scaled boundary finite-element method for elastostatics , 2002 .

[10]  R. Hauptmann,et al.  On volumetric locking of low‐order solid and solid‐shell elements for finite elastoviscoplastic deformations and selective reduced integration , 2000 .

[11]  J. Wolf,et al.  Consistent Infinitesimal Finite-Element Cell Method: Three-Dimensional Vector Wave Equation , 1996 .

[12]  Eduardo Kausel,et al.  Numerical dispersion in the thin-layer method , 2004 .

[13]  Francis Tin-Loi,et al.  Evaluation of Dynamic Generalised Stress Intensity Factors at Cracks and Multi-Material Wedges Using the Scaled Boundary Finite Element Method , 2011 .

[14]  M. Kaliske,et al.  An extended tube-model for rubber elasticity : Statistical-mechanical theory and finite element implementation , 1999 .

[15]  Paul Steinmann,et al.  Investigations on the polygonal finite element method: Constrained adaptive Delaunay tessellation and conformal interpolants , 2013 .

[16]  Chongmin Song.  Analysis of singular stress fields at multi‐material corners under thermal loading , 2006 .

[17]  Francis Tin-Loi,et al.  Scaled boundary polygons with application to fracture analysis of functionally graded materials , 2014 .

[18]  John P. Wolf,et al.  Consistent infinitesimal finite-element cell method: in-plane motion , 1995 .

[19]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis: Part II (1986-1996) , 1997 .

[20]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis , 1987 .

[21]  O. H. Yeoh,et al.  Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates , 1990 .

[22]  Chongmin Song,et al.  The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics , 1997 .

[23]  Ean Tat Ooi,et al.  Modelling dynamic crack propagation using the scaled boundary finite element method , 2011 .

[24]  Chongmin Song,et al.  Dynamic analysis of unbounded domains by a reduced set of base functions , 2006 .

[25]  Robert L. Taylor,et al.  FEAP - - A Finite Element Analysis Program , 2011 .

[26]  John P. Wolf,et al.  Semi-analytical representation of stress singularities as occurring in cracks in anisotropic multi-materials with the scaled boundary finite-element method , 2002 .

[27]  Lin Gao,et al.  Dynamic dam-reservoir interaction analysis including effect of reservoir boundary absorption , 2007 .

[28]  Nicholas J. Higham,et al.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..

[29]  Peter Wriggers,et al.  Nichtlineare Finite-Element-Methoden , 2001 .

[30]  Andrew Deeks,et al.  Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method , 2007 .

[31]  Chongmin Song.  A matrix function solution for the scaled boundary finite-element equation in statics , 2004 .

[32]  Shijun Liao,et al.  The scaled boundary FEM for nonlinear problems , 2011 .

[33]  F. Tin-Loi,et al.  Polygon scaled boundary finite elements for crack propagation modelling , 2012 .

[34]  F. Tin-Loi,et al.  A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges , 2010 .

[35]  Development of polygon elements based on the scaled boundary finite element method , 2010 .

[36]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[37]  Chongmin Song,et al.  Time‐harmonic response of non‐homogeneous elastic unbounded domains using the scaled boundary finite‐element method , 2006 .

[38]  Carolin Birk,et al.  A modified scaled boundary finite element method for three‐dimensional dynamic soil‐structure interaction in layered soil , 2012 .

[39]  E. Ooi,et al.  Efficient prediction of deterministic size effects using the scaled boundary finite element method , 2010 .

[40]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .