Stress Intensity Factor - Relationship To Energy Release Rate and J-integral

Relationship To Energy Release Rate and J-integral

The strain energy release rate for a crack under mode I loading is related to the stress intensity factor by

G = K_{\rm I}^2\left(\frac{1-\nu^2}{E}\right)

where is the Young's modulus and is the Poisson's ratio of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. Plane strain has been assumed and the crack has been assumed to extend along the direction of the initial crack. For plane stress conditions, the above relation simplifies to

G = K_{\rm I}^2\left(\frac{1}{E}\right)\,.

For pure mode II loading, we have similar relations

G = K_{\rm II}^2\left(\frac{1-\nu^2}{E}\right) \quad \text{or} \quad G = K_{\rm II}^2\left(\frac{1}{E}\right) \,.

For pure mode III loading,

G = K_{\rm III}^2\left(\frac{1}{2\mu}\right)

where is the shear modulus. For general loading in plane strain, the relationship between the strain energy and the stress intensity factors for the three modes is

G = K_{\rm I}^2\left(\frac{1-\nu^2}{E}\right) + K_{\rm II}^2\left(\frac{1-\nu^2}{E}\right) + K_{\rm III}^2\left(\frac{1}{2\mu}\right)\,.

A similar relation is obtained for plane stress by adding the contributions for the three modes.

The above relations can also be used to connect the J integral to the stress intensity factor because

G = J = \int_\Gamma \left(W~dx_2 - \mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{\partial x_1}~ds\right) \,.

Read more about this topic:  Stress Intensity Factor

Famous quotes containing the words relationship to, relationship, energy, release and/or rate:

    “Poetry is above all a concentration of the power of language, which is the power of our ultimate relationship to everything in the universe.”
    —Adrienne Rich (b. 1929)

    “The proper aim of education is to promote significant learning. Significant learning entails development. Development means successively asking broader and deeper questions of the relationship between oneself and the world. This is as true for first graders as graduate students, for fledging artists as graying accountants.”
    —Laurent A. Daloz (20th century)

    “In the west, Apollo and Dionysus strive for victory. Apollo makes the boundary lines that are civilization but that lead to convention, constraint, oppression. Dionysus is energy unbound, mad, callous, destructive, wasteful. Apollo is law, history, tradition, the dignity and safety of custom and form. Dionysus is the new, exhilarating but rude, sweeping all away to begin again. Apollo is a tyrant, Dionysus is a vandal.”
    —Camille Paglia (b. 1947)

    “As nature requires whirlwinds and cyclones to release its excessive force in a violent revolt against its own existence, so the spirit requires a demonic human being from time to time whose excessive strength rebels against the community of thought and the monotony of morality ... only by looking at those beyond its limits does humanity come to know its own utmost limits.”
    —Stefan Zweig (18811942)

    “At this very moment,... the most frightful horrors are taking place in every corner of the world. People are being crushed, slashed, disembowelled, mangled; their dead bodies rot and their eyes decay with the rest. Screams of pain and fear go pulsing through the air at the rate of eleven hundred feet per second. After travelling for three seconds they are perfectly inaudible. These are distressing facts; but do we enjoy life any the less because of them? Most certainly we do not.”
    —Aldous Huxley (1894–1963)