Three‐dimensional numerical modelling by XFEM of spring‐layer imperfect curved interfaces with applications to linearly elastic composite materials

The spring-layer interface model is widely used in describing some imperfect interfaces frequently involved in materials and structures. Typically, it is appropriate for modelling a thin soft interphase layer between two relatively stiff bulk media. According to the spring-layer interface model, the displacement vector suffers a jump across an interface whereas the traction vector is continuous across the same interface and is, in the linear case, proportional to the displacement vector jump. In the present work, an efficient three-dimensional numerical approach based on the extended finite element method is first proposed to model linear spring-layer curved imperfect interfaces and then applied to predict the effective elastic moduli of composites in which such imperfect interfaces intervene. In particular, a rigorous derivation of the linear spring-layer interface model is provided to clarify its domain of validity. The accuracy and convergence rate of the elaborated numerical approach are assessed via benchmark tests for which exact analytical solutions are available. The computated effective elastic moduli of composites are compared with the relevant analytical lower and upper bounds. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  R. Asaro.  Somigliana dislocations and internal stresses; with application to second phase hardening , 1975 .

[2]  Y. Benveniste,et al.  The effective mechanical behaviour of composite materials with imperfect contact between the constituents , 1985 .

[3]  Z. Hashin.  Extremum principles for elastic heterogenous media with imperfect interfaces and their application to bounding of effective moduli , 1992 .

[4]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[5]  Pradeep Sharma,et al.  Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities , 2003 .

[6]  F. X. Schneider,et al.  Gründung tumartiger Bauwerke: Dr. L. Varga und Dr. S. Kaliszky. 1974. Bauverlag Wiesbaden und Berlin. 253 Seiten, 126 Abbildungen, 17 Tabellen und 1 Tafel. DM 54, --. , 1976 .

[7]  Jianmin Qu,et al.  The effect of slightly weakened interfaces on the overall elastic properties of composite materials , 1993 .

[8]  G. Dvorak,et al.  Solids containing spherical nano-inclusions with interface stresses: Effective properties and thermal-mechanical connections , 2007 .

[9]  Y. Benveniste,et al.  A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media , 2006 .

[10]  Jan Drewes Achenbach,et al.  Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites , 1989 .

[11]  Huiling Duan,et al.  An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites☆ , 2005 .

[12]  Samuel Geniaut,et al.  An X‐FEM approach for large sliding contact along discontinuities , 2009 .

[13]  Toshio Mura,et al.  The stress field of a sliding inclusion , 1985 .

[14]  Ted Belytschko,et al.  The intrinsic partition of unity method , 2007 .

[15]  Zvi Hashin,et al.  The Spherical Inclusion With Imperfect Interface , 1991 .

[16]  Jean-François Remacle,et al.  A computational approach to handle complex microstructure geometries , 2003 .

[17]  G. I. Barenblatt.  THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE , 1962 .

[18]  Julien Yvonnet,et al.  An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites , 2008 .

[19]  A. Hillerborg,et al.  Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements , 1976 .

[20]  Nicolas Moës,et al.  Imposing Dirichlet boundary conditions in the extended finite element method , 2006 .

[21]  Zhu-ping Huang,et al.  Stress concentration tensors of inhomogeneities with interface effects , 2005 .

[22]  Y. Povstenko,et al.  Theoretical investigation of phenomena caused by heterogeneous surface tension in solids , 1993 .

[23]  Zvi Hashin,et al.  Thermoelastic properties of fiber composites with imperfect interface , 1990 .

[24]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[25]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[26]  M. Ortiz,et al.  FINITE-DEFORMATION IRREVERSIBLE COHESIVE ELEMENTS FOR THREE-DIMENSIONAL CRACK-PROPAGATION ANALYSIS , 1999 .

[27]  R. Jorge,et al.  On the use of element-free Galerkin Method for problems involving incompressibility , 2007 .

[28]  Nicolas Moës,et al.  A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method , 2009 .

[29]  R. Lipton,et al.  Composites with imperfect interface , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[30]  John A. Nairn,et al.  Numerical Implementation of Imperfect Interfaces , 2007 .

[31]  Bhushan Lal Karihaloo,et al.  Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress , 2005 .

[32]  Z. Hashin.  Thin interphase/imperfect interface in elasticity with application to coated fiber composites , 2002 .

[33]  Isaac Harari,et al.  A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces , 2007 .

[34]  H. Shen,et al.  Interfacial thermal stress analysis of an elliptic inclusion with a compliant interphase layer in plane elasticity , 2001 .

[35]  Z. Zhong,et al.  On the Elastic Field of a Shpherical Inhomogeneity with an Imperfectly Bonded Interface , 1997 .

[36]  T. Belytschko,et al.  On the construction of blending elements for local partition of unity enriched finite elements , 2003 .

[37]  P. Hansbo,et al.  A finite element method for the simulation of strong and weak discontinuities in solid mechanics , 2004 .

[38]  T. Fries.  A corrected XFEM approximation without problems in blending elements , 2008 .

[39]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[40]  John E. Dolbow,et al.  On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method , 2004 .

[41]  M. E. Gurtin,et al.  A general theory of curved deformable interfaces in solids at equilibrium , 1998 .