J Integrals

Use of the J-integral calculator

(This is an experimental feature, please test carefully to make sure you understand what it does.)

When solving elastic problems FEAP has a built-in feature that computes the nodal values of the J-integral; i.e. it will compute the variation of the systems potential energy with respect to variations in the nodal positions. The negative of this is the energy release rate. To use this feature, at the Macro prompt simply type JINT. To plot the J-integral vectors, enter the plot phase, and type JINT.

Note that one will see non-zero values at most nodes. This indicates that the mesh can be optimized to lower the computed energy. However, if there is a singularity in the problem, then there will be a dominant J-integral vector at the singularity that can be interpreted as the energy release rate. See

Nguyen, T.D., Govindjee, S., Klein, P.A., and Gao, H., "A Material Force Method for Inelastic Fracture Mechanics," Journal of the Mechanics and Physics of Solids, v53, 91-121 (2005).

JINT Syntax Macro phase

jint,<grad>,<n1,n2,n3> jint,all

where n1,n2 are starting and ending node numbers and n3 is an increment. If you use the grad option the formulation uses $\nabla \mathbf {u}$ under the integral sign instead of the deformation gradient $\mathbf {F}$ formulation.

Simple Example

A simple example of an edge crack

FEAP  * * Simple Test of JINT for an elastic problem * *
0 0 0 2 2 4

PARAmeter
 n = 4

MATErial 1
  SOLId
    ELAStic ISOTropic 200e3 0.3

MATErial 2
  SOLId
    ELAStic ISOTropic 200e3 0.3

MATErial 3
  SOLId
    ELAStic ISOTropic 200e3 0.3

BLOCk
 CART 4*n 8*n
 MATE 1
 1 1 -1
 2 2 -1
 3 2  1 
 4 1  1

BLOCk
 CART 4*n 4*n
 MATE 2
 1 0 0
 2 1 0
 3 1 1
 4 0 1

BLOCk
 CART 4*n 4*n
 MATE 3
 1 0 -1
 2 1 -1
 3 1  0
 4 0  0

EBOUndary
  2 -1  1 1
  2  1  1 1

EDISplacement
  2 1  0 1

END


tie,mate,1,2
tie,mate,1,3

batch
  tang,,1       ! Solve for the (linear) elastic solution
  plot outl     ! Plot mesh showing crack
  plot jint     ! Add J-integral computations to the plot
  jint,,273     ! Output J-int at crack tip node using F
  jint,grad,273 ! Output J-int at crack tip node using grad(u)
end

inte

stop