Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads
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arXiv
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| Format: | Preprint |
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2025
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| author | Ghosh, Saugata Bhatta, Dambaru Mallikarjunaiah, S. M. |
| author_facet | Ghosh, Saugata Bhatta, Dambaru Mallikarjunaiah, S. M. |
| contents | A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasilinear elliptic BVP. This BVP is solved using a robust numerical scheme combining Picard-type linearization with a continuous Galerkin finite element method for spatial discretization. Numerical results are presented for various loading conditions, including uniform tension, non-uniform slope, and parabolic loading, with two distinct material fiber orientations. It is demonstrated that crack-tip strain growth is substantially lower than stress growth. Nevertheless, strain-energy density is found to be concentrated at the crack tip, consistent with linear elastic fracture mechanics principles. The proposed framework provides a robust basis for formulating physically meaningful, rigorous BVPs, critical for investigating fundamental processes like crack propagation, damage, and nucleation in anisotropic, strain-limiting elastic solids under diverse loading conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads Ghosh, Saugata Bhatta, Dambaru Mallikarjunaiah, S. M. Numerical Analysis A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasilinear elliptic BVP. This BVP is solved using a robust numerical scheme combining Picard-type linearization with a continuous Galerkin finite element method for spatial discretization. Numerical results are presented for various loading conditions, including uniform tension, non-uniform slope, and parabolic loading, with two distinct material fiber orientations. It is demonstrated that crack-tip strain growth is substantially lower than stress growth. Nevertheless, strain-energy density is found to be concentrated at the crack tip, consistent with linear elastic fracture mechanics principles. The proposed framework provides a robust basis for formulating physically meaningful, rigorous BVPs, critical for investigating fundamental processes like crack propagation, damage, and nucleation in anisotropic, strain-limiting elastic solids under diverse loading conditions. |
| title | Computational Insights into Orthotropic Fracture: Crack-Tip Fields in Strain-Limiting Materials under Non-Uniform Loads |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.01150 |