An $hp$-adaptive finite element framework for static cracks: The impact of pointwise density variations on mode I, mode II, and mixed-mode fracture

Abstract

A robust $hp$-adaptive finite element framework is presented for the investigation of static cracks in materials characterized by complex, pointwise density variations. Within such heterogeneous media, the equilibrium equation governed by the divergence of the stress tensor is reduced to a vector-valued quasilinear partial differential equation, wherein significant gradients and nonlinearities are introduced into the governing constitutive relations by spatial density variations. To ensure that these localized field variables are accurately captured, an automated $hp$-refinement strategy is implemented so that element sizes ($h$) and polynomial degrees ($p$) are concurrently optimized. A dual-indicator approach is employed by the framework. In this approach, $h$-refinement is driven by the Kelly error estimator, which utilizes the jump of the normal derivative across element interfaces, while $p$-refinement is determined by the decay of Legendre-Fourier coefficients to assess the local smoothness of the solution. Optimal convergence rates are ensured by this dual-refinement strategy, particularly within the singular regions surrounding the crack tip where material heterogeneities and geometric discontinuities intersect. The performance of the proposed method is rigorously evaluated across three distinct loading regimes: pure tensile (Mode I), pure shear (Mode II), and mixed-mode loading. It is demonstrated through numerical simulations that the intricate interaction between the fracture process zone and the underlying material heterogeneities is effectively resolved by the $hp$-adaptive scheme. Furthermore, the influence of specific density-dependent constitutive relations on the resulting crack-tip stress fields and strain energy density distributions is analyzed in detail.