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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.06949 |
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| _version_ | 1866916784295116800 |
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| author | Ren, Huilong |
| author_facet | Ren, Huilong |
| contents | We formulate a family of scalar softening laws by setting the stored-energy density $ψ(η)=\int_{0}^η[1-F(s)]d s$, where $F$ ranges over exponential, Cauchy, logistic, half-normal, Gudermannian, hypergeometric, radical, rational, piece-wise, and rapid-decay cumulative-distribution functions (CDFs). We prove that every such law yields a degradation map that is monotone, bounded, and dissipative, rendering the associated hyperelastic material thermodynamically admissible. Working directly in spatial dimensions $d=2,3$, we establish compactness and $Γ$-convergence of the CDF-based energies to a sharp-interface Griffith functional. We further show the existence of rate-independent quasi-static evolutions by constructing global energetic solutions that satisfy both stability and energy balance. These analytical results provide a rigorous bridge between the probabilistic damage formulation and Griffith-type fracture mechanics. One illustrative example is presented to show the effectiveness of the current damage laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | CDF-Generated Damage Laws: Admissibility, Gamma-Convergence to Griffith Fracture, and Well-Posedness Ren, Huilong Analysis of PDEs Mathematical Physics We formulate a family of scalar softening laws by setting the stored-energy density $ψ(η)=\int_{0}^η[1-F(s)]d s$, where $F$ ranges over exponential, Cauchy, logistic, half-normal, Gudermannian, hypergeometric, radical, rational, piece-wise, and rapid-decay cumulative-distribution functions (CDFs). We prove that every such law yields a degradation map that is monotone, bounded, and dissipative, rendering the associated hyperelastic material thermodynamically admissible. Working directly in spatial dimensions $d=2,3$, we establish compactness and $Γ$-convergence of the CDF-based energies to a sharp-interface Griffith functional. We further show the existence of rate-independent quasi-static evolutions by constructing global energetic solutions that satisfy both stability and energy balance. These analytical results provide a rigorous bridge between the probabilistic damage formulation and Griffith-type fracture mechanics. One illustrative example is presented to show the effectiveness of the current damage laws. |
| title | CDF-Generated Damage Laws: Admissibility, Gamma-Convergence to Griffith Fracture, and Well-Posedness |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2506.06949 |