The paper by G.I. Barenblatt, "The Mathematical Theory of Equilibrium Cracks in Brittle Fracture," provides a comprehensive overview of the theory of equilibrium cracks in brittle materials. The author discusses the fundamental concepts and developments in the field, emphasizing the non-linear nature of crack propagation and the importance of considering molecular forces of cohesion near crack edges. Key points include: 1. **Introduction to the Theory**: The paper begins with an introduction to the problem of brittle fracture and the theory of cracks, highlighting the non-linear nature of crack propagation due to the expansion of cracks under load. 2. **Basic Assumptions**: The theory assumes that the body is perfectly brittle, retaining linear elasticity up to fracture. The focus is on the equilibrium of solids with cracks, where the cracks are considered as surfaces of discontinuity in displacement. 3. **Stress Intensity Factor**: The stress intensity factor \( N \) is introduced as a critical parameter, which determines the finite or infinite stress at the crack tip. The condition for finite stress at the crack tip is \( N = 0 \). 4. **Energy Release Rate**: The energy release rate \( \dot{\rho} \) is derived, which is related to the critical load for crack initiation and propagation. 5. **Development of the Theory**: The paper reviews the historical development of the theory, including the contributions of C. E. Inglis, N. I. Muskhelishvili, A. A. Griffith, and others. It highlights the importance of considering molecular forces of cohesion in the theory. 6. **Stress and Strain Near Crack Edges**: The paper analyzes the stress and strain distribution near the edges of equilibrium cracks, showing that the tensile stress at the crack tip is finite and the opposite faces of the crack close smoothly. 7. **Conclusion**: The paper concludes with a discussion on the stability of cracks and the conditions for their expansion, emphasizing the importance of considering both stable and unstable crack propagation. The paper is a foundational work in the field, providing a unified view of the basic problems in the theory of equilibrium cracks and discussing the results obtained so far.The paper by G.I. Barenblatt, "The Mathematical Theory of Equilibrium Cracks in Brittle Fracture," provides a comprehensive overview of the theory of equilibrium cracks in brittle materials. The author discusses the fundamental concepts and developments in the field, emphasizing the non-linear nature of crack propagation and the importance of considering molecular forces of cohesion near crack edges. Key points include: 1. **Introduction to the Theory**: The paper begins with an introduction to the problem of brittle fracture and the theory of cracks, highlighting the non-linear nature of crack propagation due to the expansion of cracks under load. 2. **Basic Assumptions**: The theory assumes that the body is perfectly brittle, retaining linear elasticity up to fracture. The focus is on the equilibrium of solids with cracks, where the cracks are considered as surfaces of discontinuity in displacement. 3. **Stress Intensity Factor**: The stress intensity factor \( N \) is introduced as a critical parameter, which determines the finite or infinite stress at the crack tip. The condition for finite stress at the crack tip is \( N = 0 \). 4. **Energy Release Rate**: The energy release rate \( \dot{\rho} \) is derived, which is related to the critical load for crack initiation and propagation. 5. **Development of the Theory**: The paper reviews the historical development of the theory, including the contributions of C. E. Inglis, N. I. Muskhelishvili, A. A. Griffith, and others. It highlights the importance of considering molecular forces of cohesion in the theory. 6. **Stress and Strain Near Crack Edges**: The paper analyzes the stress and strain distribution near the edges of equilibrium cracks, showing that the tensile stress at the crack tip is finite and the opposite faces of the crack close smoothly. 7. **Conclusion**: The paper concludes with a discussion on the stability of cracks and the conditions for their expansion, emphasizing the importance of considering both stable and unstable crack propagation. The paper is a foundational work in the field, providing a unified view of the basic problems in the theory of equilibrium cracks and discussing the results obtained so far.