XFEM Fracture Analysis of Composites

Nomenclature

Parameters not shown in this nomenclature are temporary variables or known constants, defined immediately when cited in the text.

α Curvilinear coordinate α First Dundurs parameter α, β Newmark parameters α, β, γ FGM constants Curvilinear coordinate α of an ellipse Components of coordinate transformation tensor β Curvilinear coordinate β Second Dundurs parameter Curvilinear coordinate β of an ellipse , Dilatational and shear wave functions γ Wedge angle Surface energy density , Dilatational and shear wave functions Engineering shear strain δ Plastic crack tip zone δ Variation of a function Dirac delta function Kronecker delta function , Local displacements of crack edges Strain tensor ε Oscillation index , Strain components Dimensionless angular geometric function Auxiliary strain components Applied displacement loading Yield strain ξ Local curvilinear (mapping) coordinate system Knot i Crack-tip position Distance function Gauss point position along the contour J η Local curvilinear (mapping) coordinate system η Equivalent inelastic strain θ Crack propagation angle with respect to initial crack θ Angular polar coordinate Crack angle , Orthotropic angular functions , Dynamic distance functions κ, Material parameters Effective material parameter λ Lame modulus λ Power of radial enrichment λ Ratio of orthotropic Young modules E2/E1 λ, Roots of the characteristic equation μ, Isotropic and orthotropic shear modulus ν, Isotropic and orthotropic Poisson's ratios Average orthotropic Poisson's ratios ρ Radius of curvature ρ Density Stress tensor Applied normal traction Critical stress for cracking von Mises effective stress , Stress components Dimensionless angular geometric function Auxiliary stress components Yield stress Hoop stress Applied tangential traction Decohesive shear stress Level set function Complex stress function Angle of orthotropic axes Crack angle Ramp function for transition domain Electric potential Enrichment function for weak discontinuities Complex stress function ψ Friction coefficient ψ Phase angle Enrichment function Level set function Complex stress function Boundary Infinitesimally small internal contour Crack boundary Traction (natural) boundary Displacement (essential) boundary Finite variation of a function Time-step Crack length increment Time interval shape functions Knot vector Π Potential energy Φ Airy stress function MLS shape functions Complex functions Ω Domain , Non-overlapping subdomains Domain associated with the partition of unity Dislocation glide enrichment (1, 2) Material axes a Crack length/half length a Semi-major axis of ellipse Effective crack length a(x) Vector of unknown coefficients a, ah Heaviside enrichment degrees of freedom ai, ak Enrichment degrees of freedom aenr Enrichment degrees of freedom a Area associated with the domain J integral A1 Area inside the infinitesimally small internal contour A+, A− Area of the influence domain above and below the crack Ai, Aij Coefficients b Width of a plate b Semi-minor axis of ellipse bk, blk Crack tip enrichment degrees of freedom Burgers vector for dislocation α Magnitude of the Burgers vector for dislocation α bn Series coefficients B Matrix of derivatives of shape functions B12, B66 Coefficients of characteristic equation B-spline basis function of order p Bh Matrix of derivatives of final shape functions Bri Strain-displacement matrix (derivatives of shape functions) Bui Matrix of derivatives of classical FE shape...