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# J integral

The J-integral represents a way to calculate work (energy) per unit fracture surface area in a material. J1c defines the point at which large-scale plastic yielding during propagation takes place under mode one loading. This value is difficult to determine experimentally, however in 1968 Jim Rice developed the J-integral test that allows one to calculate fracture toughness (K1c) for materials in which sample sizes are too small (on the order of < 1 meter) for direct determination of K1c. Physically the J-integral is related to the area under curve of a load versus load point displacement..

## J-Integral and Fracture Toughness

The J-integral can be described as follows $J=\oint_{C} \frac {F}{A}\frac{du}{dl_0}=\int_{}^{}\sigma d\varepsilon\,$

where

• F is the force applied at the crack tip
• A is the area of the crack tip
• $\frac{du}{dl_0}$ is the change in energy per unit length
• σ is the stress
• $d\varepsilon$ is the change in the strain caused by the stress

Fracture toughness is then calculated from the following equation $J_{1c} = K_{1c}^2(\frac{1-v^2}{E})$

where