The J-integral is used to calculate the energy release rate for a crack in a body subjected to monotonic loading. The original theory was developed by Rice [1]. The by far most common crack failure mode is modus I crack opening.
The purpose of J integral computation is to see if a structure with a crack of known geometry subjected to a given load is risking to fail due to uncontrolled crack propagation. The J integral computation will give a scalar value of crack release energy per unit fracture surface area. This can then be compared to JIC, which is the critical J value for uncontrolled crack propgataion. JIC is a material characteristics, which can be found by physical material testing procedure.
In contrast to stress intensity factor, K, the J integral is valid even when the material undergoes non-linear deformation.
Rice theory was extended to plane curved cracks in 3D in a thesis from Adrianus Bakker [2]. This has been adopted in LS-PrePost so LS-PrePost can handle J integral in both 2D and 3D.
The J integral in LS-PrePost has been extended further to also include effects from a residual stress field, which typically arise in welded structures. The theory is based on the work of Y. Lei et al, [3].
The assumptions and conditions required for the J-integral to be striclty valid are studied in detail in [4]
So the J integral in LS-PrePost is computed as:
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See references [2] and [3] for details.
W is the internal energy as computed from LS-DYNA and when residual stresses are included, also includes the elastic energy at t=0. For shell elements, internal energy density W is included by default in the d3plot but for solid elements, output of this quantity has to be activated with parameter HYDRO on *DATABASE_EXTENT_BINARY. In order to know where among the history variables the internal energy is stored, the user have to load the keyword file too when computing J integral for 3D models.
The theory says that the J integral should be path independent and the user can set how many elements away from the crack tip/edge will be used when computing the integral to evaluate the j integral for different contours around the crack tip/edge. This is set by the number of contours in the GUI.
The effects of residual stresses are automatically activated in the J integral computation if the loaded keyword includes *INITIAL_STRESS and *INITIAL_STRAIN. The variable W does, in the case of residual stresses, also include the elastic strain energy in the material at t=0. The elastic characteristics from the material is obtained from the keyword file.
Run the model: jintegral.key in LS-DYNA. This is a small 2D example which can be used to try out the J integral functionality. Keep in mind as this example is a symmetry model, the computed J integral value has to be multiplied by a factor of 2 to get the correct value for the full structure.
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[1] J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, Journal of Applied Mechanics, 35, 1968, pp. 379-386.
[2] Bakker Adrianus, The Three-Dimensional J-lntegral. An Investigation into Its Use for Post-Yield Fracture Safety Assessment. PhD thesis, 1984
[3] Y. Lei, N.P. O'dowd, G.A. Webster. Fracture mechanics analysis of a crack in a residual stress field, Int. Journal of Fracture, 106, 2000, pp 195-216.
[4] W. Brocks, I. Scheider. Numerical Aspects of the Path-Dependence of the J-Integral in Incremental Plasticity. Technical Note GKSS/WMS/01/08, 2001