Application of X-FEM to study two-unequal-collinear cracks in 2-D finite magnetoelectoelastic specimen R.R. Bhargava, Kuldeep Sharma ⇑ Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, India article info Article history: Received 19 December 2011 Received in revised form 1 February 2012 Accepted 9 March 2012 Available online 7 April 2012 Keywords: Electric-displacement intensity factor Extended finite element method Magnetoelectroelastic (MEE) ceramics Magnetic induction intensity factor Maximum hoop stress intensity factor Stroh formalism abstract A quasi-static and propagating analysis is simulated for two-unequal-collinear cracks in a 2-D finite mag- netoelectroelastic specimen using X-FEM. The intensity factors (IFs) are calculated using interaction inte- gral in conjugation with the near tip behavior given by the Stroh formalism. In quasi-static case energy release rate (ERR) variations are investigated with respect to inter-crack space, crack lengths and mechanical/electrical/magnetic loads. Two-collinear-unequal cracks in an infinite domain problem are simulated, analyzed and validated. Further, effects of asymmetric orientation and symmetric orientation of quasi-static two-unequal-collinear cracks vis-à-vis specimen boundaries on total energy release rate (TERR) and mechanical energy release rate (MERR) are investigated. The case of one edge and one internal quasi-static crack is obtained as a corollary and the case study is presented. Next, the crack growth study for two-unequal-collinear edge cracks is simulated using maximum modified hoop stress intensity factor criterion and considering anisotropic fracture toughness behavior of polarized ceramics. The effect of vol- ume fraction, electrical and magnetic loadings are similar to the strain energy density function criterion of crack propagation for magnetoelectroelastic ceramics. Lastly dependence of volume fraction and mate- rial constants of magnetoelectroelastic ceramics are observed on the set of standard eight basis functions for these ceramics. Hence, a more generalized set of basis functions is also defined here for these ceramics. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Magnetoelectroelastic (MEE) materials have a unique ability of converting energy from one form to the other amongst magnetic, electric and mechanical energies. These are brittle in nature and presence of imperfections or defects such as cracks, voids lead to the premature failure of these materials under mechanical, electri- cal or magnetic loadings. Thus the fracture study becomes imper- ative for such materials. Research on two-collinear-cracks problems in MEE materials got impetus in last decade. Gao et al. [1] obtained the generalized solution for collinear cracks in MEE media using Stroh-formalism subjected to permeable crack-face conditions and under arbitrary loads. Zhou et al. [2,3] employed the Schmidt method to investigate a static/dynamic behavior of two-symmetric-interface cracks be- tween two dissimilar MEE ceramics half-planes under anti- plane shear stress loads. Zhou et al. [4] also studied mode-I crack prob- lem of two-collinear permeable cracks in a MEE composite mate- rial plane using the generalized Almansi’s theorem in conjugation with Schmidt method. Wang et al. [5] studied a periodic array of cracks in a transversely isotropic MEE material using Hankel trans- forms considering both permeable and impermeable crack-face conditions. They further investigated effect of crack-spacing on intensity factors, stress, electric displacement and magnetic induc- tion. Singh et al. [6] derived closed-form solutions for anti-plane problem of two-collinear-cracks in a MEE layer of finite thickness under different boundary conditions. Zhong [7] simulated semi-permeable crack-face boundary conditions for a case of two-collinear-cracks full of a dielectric interior in a MEE ceramic. Rojas-Diaz et al. [8] analyzed dynamic interactions amongst cracks embedded in 2-D piezoelectric–piezomagnetic composite material by means of boundary element method. Li et al. [9] addressed the anti-plane case of collinear unequal crack series in MEE materials under impermeable conditions. Li and Lee [10] proposed a real fundamental solution for two- unequal-collinear cracks in MEE materials under mode-I conditions. However, for practical applications and study of fracture test on finite specimens, complex geometry, typical electromechanical boundary conditions and material non-linearity analysis require numerical techniques such as finite element method (FEM), boundary element method (BEM), mesh free method and extended finite element method (X-FEM). Sanchez et al. [11] applied the BEM to study fracture behavior of MEE composite materials. They further compared the accuracy of boundary element solution with 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.013 ⇑ Corresponding author. Tel.: +91 9456318765. E-mail address: kuldeeppurc@gmail.com (K. Sharma). Computational Materials Science 60 (2012) 75–98 Contents lists available at SciVerse ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci