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:

    “Film music should have the same relationship to the film drama that somebody’s piano playing in my living room has to the book I am reading.”
    —Igor Stravinsky (1882–1971)

    “When any relationship is characterized by difference, particularly a disparity in power, there remains a tendency to model it on the parent-child-relationship. Even protectiveness and benevolence toward the poor, toward minorities, and especially toward women have involved equating them with children.”
    —Mary Catherine Bateson (20th century)

    “I say, stamping the words with emphasis,
    Drink from here energy and only energy,”
    —Stephen Spender (1909–1995)

    “Come, thou long-expected Jesus,
    born to set thy people free;
    from our fears and sins release us,
    let us find our rest in thee.”
    —Charles Wesley (1707–1788)

    “Put God in your debt. Every stroke shall be repaid. The longer the payment is withholden, the better for you; for compound interest on compound interest is the rate and usage of this exchequer.”
    —Ralph Waldo Emerson (1803–1882)