Math 2.0

Mechanics of deformable solids Definition

J-integral

Builds on Strain energy density functionCauchy stress tensorHyperelastic material

Definition

Let a two-dimensional body be made of a homogeneous hyperelastic material with strain energy density WW. Let it contain a straight crack along the negative x1x_1-axis with tip at the origin. Let uu be the displacement field, σ\sigma the stress, and assume there are no body forces and the crack faces are traction-free. Let Γ\Gamma be a piecewise smooth curve from the lower crack face to the upper crack face, going counterclockwise around the tip. Let nn be its outer unit normal and ss its arc length. The J-integral along Γ\Gamma is J=∫Γ(Wn1−σijnj∂ui∂x1)ds,J = \int_{\Gamma} \left( W n_1 - \sigma_{ij} n_j \frac{\partial u_i}{\partial x_1} \right) ds, with summation over i,j∈{1,2}i, j \in \set{1, 2}.

Remarks

Under these hypotheses JJ does not depend on the choice of Γ\Gamma. This follows from the divergence theorem: the integrand is the flux of the first row of the Eshelby energy-momentum tensor, which is divergence-free. Example: take a linear elastic isotropic body in mode I, and evaluate JJ on a small circle around the tip with the leading crack-tip fields. This gives J=KI2/E′J = K_I^2 / E', which is the energy release rate. Non-example: if the crack faces carry tractions, or the material varies with x1x_1, the integral depends on the contour. Its value on a given contour is then not a J-integral in the sense above. For elastic–plastic materials the integral is evaluated with the deformation theory of plasticity, which is formally a nonlinear elastic theory. So path independence holds as long as there is no unloading.

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  1. Cauchy stress tensor concept
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  4. J-integral definition

Further reading

  1. Web https://en.wikipedia.org/wiki/J-integral
About this page
TagWJ8EBVY08S
URIm2://WJ8EBVY08S
Kinddefinition
WritingW0 stub
MSC74R10, 74R20, 74A45
DefinesJ-integral
UsesStrain energy density function, Cauchy stress tensor, Hyperelastic material
Changed3 November 2026

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