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:

    “... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
    —Adrienne Rich (b. 1929)

    “Our mother gives us our earliest lessons in love—and its partner, hate. Our father—our “second other”Melaborates on them. Offering us an alternative to the mother-baby relationship . . . presenting a masculine model which can supplement and contrast with the feminine. And providing us with further and perhaps quite different meanings of lovable and loving and being loved.”
    —Judith Viorst (20th century)

    “Because humans are not alone in exhibiting such behavior—bees stockpile royal jelly, birds feather their nests, mice shred paper—it’s possible that a pregnant woman who scrubs her house from floor to ceiling [just before her baby is born] is responding to a biological imperative . . . . Of course there are those who believe that . . . the burst of energy that propels a pregnant woman to clean her house is a perfectly natural response to their mother’s impending visit.”
    —Mary Arrigo (20th century)

    “The shallow consider liberty a release from all law, from every constraint. The wise man sees in it, on the contrary, the potent Law of Laws.”
    —Walt Whitman (1819–1892)

    “All of us failed to match our dreams of perfection. So I rate us on the basis of our splendid failure to do the impossible.”
    —William Faulkner (1897–1962)