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:

    “Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.”
    —Johann Wolfgang Von Goethe (1749–1832)

    “Most childhood problems don’t result from “bad” parenting, but are the inevitable result of the growing that parents and children do together. The point isn’t to head off these problems or find ways around them, but rather to work through them together and in doing so to develop a relationship of mutual trust to rely on when the next problem comes along.”
    —Fred Rogers (20th century)

    “The principle of avoiding the unnecessary expenditure of energy has enabled the species to survive in a world full of stimuli; but it prevents the survival of the aristocracy.”
    —Rebecca West (1892–1983)

    “An inquiry about the attitude towards the release of so-called political prisoners. I should be very sorry to see the United States holding anyone in confinement on account of any opinion that that person might hold. It is a fundamental tenet of our institutions that people have a right to believe what they want to believe and hold such opinions as they want to hold without having to answer to anyone for their private opinion.”
    —Calvin Coolidge (1872–1933)

    ““Terence, this is stupid stuff:
    You eat your victuals fast enough;
    There can’t be much amiss, ‘tis clear,
    To see the rate you drink your beer.”
    —A.E. (Alfred Edward)