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A finite element model to analyze crack-tip fields in a transversely isotropic strain-limiting elastic solid

Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

TL;DR

The paper addresses crack-tip fields in transversely isotropic, strain-limiting elastic solids by adopting Rajagopal's implicit elasticity to impose a bounded-strain response near singularities. A vector-valued, quasi-linear elliptic boundary-value problem is formulated with $\boldsymbol{\epsilon}=\boldsymbol{F}(\boldsymbol{T})$ and its hyperelastic inversion $\boldsymbol{T}=\Psi(\|\boldsymbol{E}^{1/2}[\boldsymbol{\epsilon}]\|)\boldsymbol{E}[\boldsymbol{\epsilon}]$, and solved via a Picard-type linearization within a continuous Galerkin finite-element framework. The authors prove well-posedness of the continuous and discrete weak formulations, establish Lipschitz and monotonicity properties of the nonlinear operator, and demonstrate convergence of the nonlinear solver. Numerical results for two fiber orientations show that crack-tip strains grow more slowly than stresses and that the strain-energy density remains concentrated near the tip, with elliptical crack-opening profiles, indicating a physically meaningful regularization of tip fields and the potential to study crack evolution and damage in anisotropic materials within this nonlinear framework.

Abstract

This paper presents a finite element model for the analysis of crack-tip fields in a transversely isotropic strain-limiting elastic body. A nonlinear constitutive relationship between stress and linearized strain characterizes the material response. This algebraically nonlinear relationship is critical as it mitigates the physically inconsistent strain singularities that arise at crack tips. These strain-limiting relationships ensure that strains remain bounded near the crack tip, representing a significant advancement in the formulation of boundary value problems (BVPs) within the context of first-order approximate constitutive models. For a transversely isotropic elastic material containing a crack, the equilibrium equation, derived from the balance of linear momentum under a specified nonlinear constitutive relation, is shown to reduce to a second-order, vector-valued, quasilinear elliptic BVP. A robust numerical method is introduced, integrating Picard-type linearization with a continuous Galerkin-type finite element procedure for spatial discretization. Numerical results, obtained for tensile loading conditions and two distinct material fiber orientations, illustrate that the evolution of crack-tip strains occurs significantly slower than that of the normalized stresses. However, the strain-energy density is most pronounced near the crack tip, consistent with observations from linearized elasticity theory. It is demonstrated that the framework investigated herein can serve as a basis for formulating physically meaningful and mathematically well-defined BVPs, which are essential for exploring crack evolution, damage, nucleation, and failure in anisotropic strain-limiting elastic materials.

A finite element model to analyze crack-tip fields in a transversely isotropic strain-limiting elastic solid

TL;DR

The paper addresses crack-tip fields in transversely isotropic, strain-limiting elastic solids by adopting Rajagopal's implicit elasticity to impose a bounded-strain response near singularities. A vector-valued, quasi-linear elliptic boundary-value problem is formulated with and its hyperelastic inversion , and solved via a Picard-type linearization within a continuous Galerkin finite-element framework. The authors prove well-posedness of the continuous and discrete weak formulations, establish Lipschitz and monotonicity properties of the nonlinear operator, and demonstrate convergence of the nonlinear solver. Numerical results for two fiber orientations show that crack-tip strains grow more slowly than stresses and that the strain-energy density remains concentrated near the tip, with elliptical crack-opening profiles, indicating a physically meaningful regularization of tip fields and the potential to study crack evolution and damage in anisotropic materials within this nonlinear framework.

Abstract

This paper presents a finite element model for the analysis of crack-tip fields in a transversely isotropic strain-limiting elastic body. A nonlinear constitutive relationship between stress and linearized strain characterizes the material response. This algebraically nonlinear relationship is critical as it mitigates the physically inconsistent strain singularities that arise at crack tips. These strain-limiting relationships ensure that strains remain bounded near the crack tip, representing a significant advancement in the formulation of boundary value problems (BVPs) within the context of first-order approximate constitutive models. For a transversely isotropic elastic material containing a crack, the equilibrium equation, derived from the balance of linear momentum under a specified nonlinear constitutive relation, is shown to reduce to a second-order, vector-valued, quasilinear elliptic BVP. A robust numerical method is introduced, integrating Picard-type linearization with a continuous Galerkin-type finite element procedure for spatial discretization. Numerical results, obtained for tensile loading conditions and two distinct material fiber orientations, illustrate that the evolution of crack-tip strains occurs significantly slower than that of the normalized stresses. However, the strain-energy density is most pronounced near the crack tip, consistent with observations from linearized elasticity theory. It is demonstrated that the framework investigated herein can serve as a basis for formulating physically meaningful and mathematically well-defined BVPs, which are essential for exploring crack evolution, damage, nucleation, and failure in anisotropic strain-limiting elastic materials.

Paper Structure

This paper contains 11 sections, 5 theorems, 22 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Implicit theory of elasticity
  4. Boundary value problem and existence of solution
  5. Galerkin-type finite element method
  6. Continuous weak formulation
  7. Discrete finite element formulation
  8. Numerical results and discussion
  9. Fiber orientation along $x$-axis
  10. Fiber orientation perpendicular to $x$-axis
  11. Conclusion

Key Result

Theorem 3.1

Consider a bounded, connected, Lipschitz domain $\mathcal{D} \subset \mathbb{R}^2$ with a partitioned Lipschitz boundary. The boundary is composed of a relatively open Dirichlet boundary $\Gamma_D$ and a relatively open Neumann boundary $\Gamma_N$, such that $\Gamma_D \, \cap \, \Gamma_N = \emptyset Weak Formulation and Existence: The existence of a solution pair $(\boldsymbol{u}, \, \boldsymbol{T

Figures (9)

  • Figure 1: Computational domain
  • Figure 2: Strain plots for different parameter variations.
  • Figure 3: Stress plots for different parameter variations.
  • Figure 4: Energy density plots for different parameter variations.
  • Figure 5: Displacement plots for $\alpha = 1.0$, $\sigma_{T} = 0.1$, and $\beta = 10.0$ in various configurations.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Lemma 3.2
  • Remark 1
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3