A computational model for crack-tip fields in a 3-D porous elastic solid with material moduli dependent on density

TL;DR

This work develops a 3-D crack-tip model for porous elastic solids whose material moduli depend on density, adopting an implicit constitutive framework in which $0=\beta_0\boldsymbol{I}+\beta_1\boldsymbol{T}+\beta_2\boldsymbol{B}+\dots$ and density effects enter via parameters like $\beta$. The static BVP is solved with a continuous Galerkin finite-element method and Picard linearization, yielding strain-limiting behavior for negative $\beta$ while preserving classical fracture senses such as the stress intensity factor $K_I$ and strain energy density concentration near the tip. Computational results on a 3-D plate with a V-notch show that negative $\beta$ reduces crack-tip strain and increases tip stress, whereas positive $\beta$ has the opposite effect, with near-tip $K_I$ remaining close to linear-elastic predictions. The framework enables fracture analysis in density-dependent porous materials without unphysical singularities and sets the stage for extensions such as phase-field regularization and surface mechanics to further regularize crack-tip fields.

Abstract

A mathematical model for crack-tip fields is proposed in this paper for the response of a three-dimensional (3-D) porous elastic solid whose material moduli are dependent on the density. Such a description wherein the generalized Lamè coefficients are nonlinear functions of material stiffness is more realistic because most engineering materials are porous, and their material properties depend on porosity and density. The governing boundary value problem for the static equilibrium state in a 3-D, homogeneous, isotropic material is obtained as a second-order, quasilinear partial-differential-equation system with a classical traction-free crack-surface boundary condition. The numerical solution is obtained from a continuous trilinear Galerkin-type finite element discretization. A Picard-type linearization is utilized to handle the nonlinearities in the discrete problem. The proposed model can describe the state of stress and strain in various materials, including recovering the classical singularities in the linearized model. The role of \textit{tensile stress}, \textit{stress intensity factor} (SIF), and \textit{strain energy density} are examined. The results indicate that the maximum values of all these quantities occur directly before the crack-tip, consistent with the observation made in the canonical problem for the linearized elastic fracture mechanics. One can use the same classical local fracture criterion, like the maximum of SIF, to study crack tips' quasi-static and dynamic evolution within the framework described in this article.

Figures (8)

• Figure 1: The geometry of the crack. The crack is on a square plate. The crack angle is $2^o$. The $xyz$ rectangular coordinate system is displayed on the geometry.
• Figure 2: The tensile displacement boundary conditions are applied on two sides of the plate parallel to the $x$-axis. Other boundaries are traction-free. Tetrahedral meshes are prescribed over the cracked plate. To produce more accurate computational results, finer meshes are created near the crack tip.
• Figure 3: Illustration of $\epsilon_{22}$ over the whole geometrical surface and discrete slices in different directions. All the panels complement each other to facilitate understanding of the $\epsilon_{22}$ distribution. The concentration of $\epsilon_{22}$ forming two branches near the crack tip can be viewed particularly in Figs. \ref{['surffa']} and \ref{['xyslice']}.
• Figure 4: $\epsilon_{22}$ vs. $r$ for different $\beta$ values. For negative $\beta$, greater $|\beta|$ produces smaller $\epsilon_{22}$ at the crack-tip ($r=0$), realizing the strain-limiting effect. On the contrary, for positive $\beta$, greater $|\beta|$ produces greater $\epsilon_{22}$ at the crack-tip ($r=0$), failing to realize the strain-limit effect. All curves for both positive and negative $\beta$ values show slightly wavy behaviors.
• Figure 5: A transparent surface view of $T_{22}$ distribution (unit: $10^4$ Pa) at $\beta=-30$. The concentration of $T_{22}$ can be viewed near the crack tip and on the rear part of the left and right lateral sides. The general pattern is similar to the one for $\epsilon_{22}$ in Fig. \ref{['surffa']}.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4