Analysis Of Stresses And Strains Near The End Of A Crack Traversing A Plate Irwinl

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Analysis Of Stresses And Strains Near The End Of A Crack Traversing A Plate Irwinl: A Classic Paper In Fracture Mechanics

In 1957, George R. Irwin published a seminal paper in the Journal of Applied Mechanics titled \"Analysis Of Stresses And Strains Near The End Of A Crack Traversing A Plate\". This paper introduced the concept of stress intensity factor, a key parameter in fracture mechanics that characterizes the severity of a crack tip singularity. Irwin also derived the famous Westergaard solution for the stress and displacement fields near a crack tip in an infinite elastic plate subjected to remote tension.

The paper has been cited over 10,000 times and has influenced many subsequent developments in fracture mechanics, such as the crack tip opening displacement (CTOD) criterion, the J-integral, the strain energy release rate (SERR), and the finite element method (FEM). In this article, we will review the main contributions of Irwin's paper and discuss its relevance and applications in modern engineering.

Stress Intensity Factor

One of the main challenges in fracture mechanics is to predict the onset and propagation of cracks in materials and structures under various loading conditions. To do this, we need to know how the stress and strain fields vary near the crack tip, where the material experiences the highest stress concentration and deformation. However, solving the elasticity equations for a cracked body is not trivial, especially for complex geometries and boundary conditions.

Irwin proposed a simple but powerful way to characterize the crack tip singularity by introducing a dimensionless parameter called the stress intensity factor (SIF), denoted by K. The SIF is defined as the coefficient of the leading term in the asymptotic expansion of the stress field near the crack tip. For example, for a mode I (opening) crack in an infinite plate under remote tension Ï, Irwin showed that the stress field near the crack tip can be expressed as:

$$\\sigma_{ij} = \\frac{K_I}{\\sqrt{2\\pi r}}f_{ij}(\\theta)+O(r^{-1/2})$$

where r and Î are polar coordinates centered at the crack tip, K_I is the mode I SIF, and f_ij(Î) are dimensionless functions that depend on the crack geometry and loading mode. The SIF can be interpreted as a measure of the intensity of the stress field near the crack tip. A higher SIF means a higher stress concentration and a higher tendency for crack growth.

The SIF also has a physical meaning in terms of energy. Irwin showed that the SIF is proportional to the square root of the strain energy release rate (SERR), which is the amount of energy released per unit area of crack growth. The SERR can be used to determine the fracture toughness of a material, which is the critical value of SERR or SIF required to initiate crack growth. Irwin also established a relation between the SIF and the crack tip opening displacement (CTOD), which is another important parameter in fracture mechanics that describes how much the crack faces separate near the tip.

Westergaard Solution

Another major contribution of Irwin's paper was to derive an exact solution for the stress and displacement fields near a crack tip in an infinite elastic plate subjected to remote tension. This solution was originally obtained by Westergaard in 1939 using complex variable methods, but Irwin presented a simpler derivation using Cartesian coordinates and Airy stress functions.

The Westergaard solution is given by:

$$\\sigma_{xx} = \\frac{K_I}{\\sqrt{2\\pi r}}\\cos\\frac{\\theta}{2}\\left(1-\\sin\\frac{\\theta}{2}\\sin\\frac{3\\theta}{2}\\right)$$

$$\\sigma_{yy} = \\frac{K_I}{\\sqrt{2\\pi r}}\\cos\\frac{\\theta}{2}\\left(1+\\sin\\frac{\\theta}{2}\\sin\\frac{3\\theta}{2}\\right)$$

$$\\sigma_{xy} = \\frac{K_I}{\\sqrt{2\\pi r}}\\cos\\frac{\\theta}{ 061ffe29dd