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problem about the Jacobian matrix (DDSDDE) in UMAT

Hi all friends,

I have a question. Why the Jacobian matrix (DDSDDE) in UMAT is defined as:


rather than:


And can anybody tell me what's the difference between them?


Thank you all for your help,

Tianhao Yan


Dear YTH

It is bbecause ABAQUS solves the problems using incremental approache.

Thans afallah!

After two years, I would like to add a comment about my new understanding of this question. 

∂σ/∂ε is the tangent stiffness, which is not consistent with the time integration algorithm. That means it cannot achieve the quadratic convergence in the Newton-Raphson iteration.

∂Δσ/∂Δε is the consistent tangent stiffness, which consistent with the time integration algorithm where the finite time increment is used.

Details can be referred to Simo and Taylor (1985)

In constituitive models without time intergation, eg. (nonlinear) elasticity, these two are essentially the same. In contrast, for plasticity, viscoelasticity, viscoplasticity, etc, which with time integration, the two tangent stiffness are different, and the consistent tangent stiffness should be used.


Simo JC, Taylor RL. Consistent tangent operator for rate-independent elastoplasticity. Comput Methods Appl Mech Eng. 1985;48:101–18.  

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