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:

    “Sometimes in our relationship to another human being the proper balance of friendship is restored when we put a few grains of impropriety onto our own side of the scale.”
    —Friedrich Nietzsche (1844–1900)

    “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 very presence of guilt, let alone its tenacity, implies imbalance: Something, we suspect, is getting more of our energy than warrants, at the expense of something else, we suspect, that deserves more of our energy than we’re giving.”
    —Melinda M. Marshall (20th century)

    “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)

    “We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
    —Max Lerner (b. 1902)