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The J-integral represents a way to calculate the strain energy release rate, or work per unit fracture surface area, in a material. [1] The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov [2] and independently in 1968 by James R. Rice, [3] who showed that an energetic contour path integral (called J) was independent of the path around a crack.
In fracture mechanics, the energy release rate, , is the rate at which energy is transformed as a material undergoes fracture.Mathematically, the energy release rate is expressed as the decrease in total potential energy per increase in fracture surface area, [1] [2] and is thus expressed in terms of energy per unit area.
the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture. the dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces that may be at work). The dissipated energy provides the thermodynamic resistance to fracture.
The fracture toughness and the critical strain energy release rate for plane stress are related by = where is the Young's modulus. If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.
This allows the strain energy release rate, , to be defined by the critical crack opening displacement, = or the critical cohesive zone size, , as follows: [6] G c = 2 ∫ 0 ν c σ y y d ν = 8 σ t h 2 r c o π E = 2 γ s {\displaystyle G_{c}=2\int _{0}^{\nu _{c}}\sigma _{yy}d\nu ={\frac {8\sigma _{th}^{2}r_{co}}{\pi E}}=2\gamma _{s}}
Using the compliance method, the critical strain energy release rate is given by G I c = 3 P C δ C 2 B a {\displaystyle G_{Ic}={\frac {3P_{C}\delta _{C}}{2Ba}}} (2) where P C {\displaystyle P_{C}} and δ C {\displaystyle \delta _{C}} are the maximum load and displacement respectively by determining when the load deflection curve has become ...
Strain energy release rate per unit fracture surface area is calculated by J-integral method which is a contour path integral around the crack tip where the path begins and ends on either crack surfaces. J-toughness value signifies the resistance of the material in terms of amount of stress energy required for a crack to grow.
The critical parameter in fracture mechanics is the stress intensity factor (K), which is related to the strain energy release rate (G) and the fracture toughness (G c). When the stress intensity factor reaches the material's fracture toughness, crack propagation becomes unstable, leading to failure.