Calculation of Deformation Gradient

[K.Li]

Dear FEAP users,

I have a general question about the calculation of deformation gradient F in FEAP. I know one can find F by the displacements of all the nodes and shape functions which make the displacement field smooth (continuous). But how the displacement of each node was calculated? It is easy to understand if it is a displacement driven boundary condition. What about a load driven simulation? So if we know the deformation gradient from last simulation step, then how did FEAP determine the deformation gradient for next step? Does it depend on the deformation gradient and stress output from last step only? What else?  Any references on this topic?

Thanks.

[leifamer]

I have a general question about the calculation of deformation gradient F in FEAP. I know one can find F by the displacements of all the nodes and shape functions which make the displacement field smooth (continuous). But how the displacement of each node was calculated? It is easy to understand if it is a displacement driven boundary condition. What about a load driven simulation? So if we know the deformation gradient from last simulation step, then how did FEAP determine the deformation gradient for next step? Does it depend on the deformation gradient and stress output from last step only? What else?  Any references on this topic?

Thanks.

Displacement is c0 continuous, but it has nothing to do with F. From current configuration you will get F with some computation. You have to learn more about nonlinear continuums theory. Read the FEAP theory manuel. 

[K.Li]

 

Displacement is c0 continuous, but it has nothing to do with F. From current configuration you will get F with some computation. You have to learn more about nonlinear continuums theory. Read the FEAP theory manuel. 

Hello, leifamer:

Thank you for your reply! But I am not sure what do you mean. At first, as I mentioned in my original question, the displacement field of elements can only be c0 continuous with the help of shape functions. Because displacement of each node within one element does not have to be c0 continuous with respect to another (no position-wise c0 continuous). The continuous displacement was obtained by interpolation of all the nodes. Also if you have 10 time steps, at each time step you have three components of displacement for each node. So time-wise it cannot be "c0 continuous" neither, and it should not be. Secondly, displacement do have something to do with deformation gradient. Check your "FEAP theory manuel" on page 106 as it states "The deformation gradient may be expressed in terms of the displacement as " in eq (13.12). My original question was about how to determine current deformation gradient (t_n) from previous time step, t_{n-1}. I just want to know a general idea whether it is calculated based on previous deformation gradient and stress output. I do know how to calculate deformation gradient if I knew the "current configuration". Surely, I am always willing to learn more "nonlinear continuums theory". 

[leifamer]

OK, to your question, i think, the current deformation F(tn)  depends only on current displacement (time tn), not the previous time step, not any stresses, as the equation F=1+u/X tells you. 

for time step tn.
F(tn) = 1+ u(tn) / X

[K.Li]

OK, to your question, i think, the current deformation F(tn)  depends only on current displacement (time tn), not the previous time step, not any stresses, as the equation F=1+u/X tells you. 

for time step tn.
F(tn) = 1+ u(tn) / X

Again, thank you for all your answers! As I mentioned before, I do know how to compute deformation gradient if I knew the current configuration which means the current coordinates of all the nodes. My original question was about how to determine current deformation gradient, F(t_n), from previous time step, t_{n-1} during a FE simulation with many time steps, and it should somehow relate to the material responses in previous step.

[leifamer]

Maybe you are doing some time discontinuous calculation?
Sorry for misunderstanding...

[Prof. R.L. Taylor]

The deformation gradient is computed from the best estimate for the solution at what you call t_n.  At the first global solution the best estimate is taken as the solution at t_n-1.  As the iterations converge, then the best estimate converges to the new solution.  The solution is driven by two factors, change in loading (either mechanical like forces or thermal or whatever non-mechanical effect is changing); or by changes in the boundary displacements.  There is no difference in computing the deformation gradient than in computing the strain.  At the beginning of a solution step one does not know the final strain, but after solution it is easily computed.

[K.Li]

Thank you for the information, Prof Taylor!