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

## Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate

By G. R. Irwin

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Published in 1957 in the Journal of Applied Mechanics

## Abstract

A substantial fraction of the mysteries associated with crack extension might be eliminated if the description of fracture experiments could include some reasonable estimate of the stress conditions near the leading edge of a crack particularly at points of onset of rapid fracture and at points of fracture arrest. It is pointed out that for somewhat brittle tensile fractures in situations such that a generalized plane-stress or a plane-strain analysis is appropriate, the influence of the test configuration, loads, and crack length upon the stresses near an end of the crack may be expressed in terms of two parameters. One of these is an adjustable uniform stress parallel to the direction of a crack extension. It is shown that the other parameter, called the stress-intensity factor, is proportional to the square root of the force tending to cause crack extension. Both factors have a clear interpretation and field of usefulness in investigations of brittle-fracture mechanics.

## Introduction

The problem of determining the stresses and strains near the end of a crack traversing a plate has been studied by many authors, such as Griffith, Inglis, Westergaard, Williams, and others. The main difficulty in solving this problem is that the stress field near the crack tip is singular, meaning that it becomes infinite as the distance from the tip approaches zero. This singularity makes it impossible to apply the usual methods of elasticity theory, such as superposition, boundary conditions, or compatibility equations. Therefore, special techniques have been developed to deal with this singularity, such as complex variable methods, asymptotic expansions, or numerical methods.

The purpose of this article is to present a simple and practical way of analyzing the stresses and strains near the end of a crack traversing a plate, based on the concept of stress-intensity factor introduced by Irwin. The stress-intensity factor is a measure of the intensity of the stress field near the crack tip, and it depends on the geometry of the crack, the applied loads, and the material properties. Irwin showed that for brittle materials, such as metals or ceramics, the fracture occurs when the stress-intensity factor reaches a critical value, which is characteristic of the material. This critical value is called the fracture toughness, and it can be determined experimentally by testing specimens with different crack lengths under different loading conditions.

## Theory

Consider a plate with thickness b and width W containing a crack with length 2a as shown in Figure 1. The plate is subjected to a uniform tensile stress σ at infinity along the x-axis. The crack lies along the y-axis and its tip is at the origin. We assume that plane-stress or plane-strain conditions are applicable, depending on whether b is small or large compared to W and 2a.

Figure 1: A plate with a crack under uniform tension

The stress field near the crack tip can be expressed in terms of polar coordinates (r, θ) as follows:

σrr = KI/(2πr) cos(θ/2) (1 + sin(θ/2) sin(3θ/2)) + σ σθθ = KI/(2πr) cos(θ/2) (1 - sin(θ/2) sin(3θ/2)) + σ σrθ = KI/(2πr) sin(θ/2) cos(θ/2) cos(3θ/2) εrr = -KI/E(2πr) cos(θ/2) (1 - ν + sin(θ/2) sin(3θ/2)) - σ/E εθθ = -KI/E(2πr) cos(θ/2) (1 - ν - sin(θ/2) sin(3θ/2)) - σ/E εrθ = -KI/E(2πr) sin(θ/2) cos(θ/2) cos(3θ/2)

where σrr, σθθ, and σrθ are the normal and shear stresses, εrr, εθθ, and εrθ are the normal and shear strains, E is the Young's modulus, ν is the Poisson's ratio, and KI is the stress-intensity factor. The stress-intensity factor is given by:

KI = σ(πa) F(a/W)

where F(a/W) is a dimensionless function of the crack length to width ratio, which can be obtained from tables or charts for various crack configurations. For example, for a center-cracked plate with 2a/W

The stress-intensity factor KI represents the magnitude of the stress singularity near the crack tip, and it is proportional to the square root of the force per unit thickness acting on the crack faces. The stress field near the crack tip is independent of the crack length and the applied stress, as long as KI remains constant. Therefore, KI can be used as a parameter to characterize the state of stress near the crack tip for different loading conditions and crack geometries.

## Fracture Criterion

The fracture criterion proposed by Irwin is based on the assumption that brittle materials fail when the maximum tensile stress near the crack tip reaches a critical value σc, which is a material property. The maximum tensile stress occurs at θ = 0 and r = rc, where rc is a small distance from the crack tip that depends on the microstructure of the material. Therefore, the fracture criterion can be written as:

KIc/(2πrc) + σ = σc

where KIc is the critical value of the stress-intensity factor, which is also called the fracture toughness. The fracture toughness can be determined experimentally by testing specimens with different crack lengths under different loading conditions, and measuring the load at which fracture occurs. The fracture toughness is a measure of the resistance of a material to crack propagation, and it depends on factors such as temperature, loading rate, environment, and microstructure.

## Conclusion

The analysis of stresses and strains near the end of a crack traversing a plate by Irwin provides a simple and practical way of characterizing the state of stress near the crack tip for different loading conditions and crack geometries. The concept of stress-intensity factor allows to compare the fracture behavior of different materials and to predict the fracture load for a given crack configuration. The fracture criterion based on the critical value of the stress-intensity factor or the fracture toughness gives a quantitative measure of the resistance of a material to crack propagation.

## References

G. R. Irwin, Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate, Journal of Applied Mechanics, Vol. 24, No. 3, pp. 361-364, 1957.

A. A. Griffith, The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of the Royal Society A, Vol. 221, No. 582-593, pp. 163-198, 1921.

Transactions of the Institute of Naval Architects, Vol. 55, pp. 219-241, 1913.

H. M. Westergaard, Bearing Pressures and Cracks, Journal of Applied Mechanics, Vol. 6, No. 1, pp. A49-A53, 1939.

M. L. Williams, On the Stress Distribution at the Base of a Stationary Crack, Journal of Applied Mechanics, Vol. 24, No. 1, pp. 109-114, 1957.