The first variation of a strain energy  usually provides the weak form (equilibrium) the second variation is related to the computation of the tangent matrix. The variation is usually computed using the differentiation, e.g. for the Green strain tensor E = 1/2 (F^T F-1) the first variation is dE = 1/2(dF^T F + F^T dF) where d() is for the deformation gradient F

d/de (F(u+e du))]e=0 = d/de (Grad(u+e du))]e=0 = Grad du

that means that you have to compute a directional derivative. The same is true for the second variation. There you have to compute the directional derivative with respect to a Delta u.

If you have rotations involved this needs more effort due to the rotation tensor. You might find a good description in

Geometrically exact analysis of spatial framesPM Pimenta, T Yojo

Appl Mech Rev 46 (11), 118-128

or

A triangular finite shell element based on a fully nonlinear shell formulationEMB Campello, PM Pimenta, P Wriggers

Computational Mechanics 31 (6), 505-518 

for beams and shells.

Regards

Peter Wriggers

Professor Wriggers's reply is deeply appreciated. The recommended literature really helps a lot to get the hang of this "second variation". Thank you! The problem challenging me involves large rotation indeed. Although it seems commonplace that compound rotations can't be added as vectors, there is several publications saying with their proposed formulation, the update of rotation parameter is additive. See Adnan Ibrahimbegović 1995, Battini, Jean Marc 2002 to name a few of them. This additive property is really alluring but i don't see popular application of those method. What is the technology they have utilized to achieve such a property? Is there any  weak point of them? 

http://www.sciencedirect.com/science/article/pii/S0045782501003528

http://onlinelibrary.wiley.com/doi/10.1002/nme.1620382107

Article Co-rotational beam elements with warping effects in stability problems

Article Computational aspects of vector-like parametrization of thre...

Dear G.M.

"Variation" is nothing but a differential/derivative:

of first order is when one calculates the first differential/gradient of a function;

of second order is when one calculates its second differential/Hessian, etc.

I am sure you are familiar with all this.

However, the way the Abaqus documentation is written is far from helpful (and this extends to many other commercial FE packages). I would recommend instead a few good books where "linearization" is thoroughly discussed:

Nonlinear Finite Elements for Continua and Structures

Ted Belytschko, Wing Kam Liu, Brian Moran

Computational Methods for Plasticity: Theory and Applications

Eduardo A. de Souza Neto, Djordje Peric, David R. J. Owen

Nonlinear Continuum Mechanics for Finite Element Analysis

Bonet, Wood

Finally:

- rotations can be "added" only when their axes coincide;

- however, the rates of arbitrary rotations can always be composed.

Best,

Stefan

Dear Stefen,

Thanks for your answer. Second variation is now understandable to me. About finite rotation, is there any parameterization strategy (in light of rotation matrix's orthogonality property, one can use 3 parameters to characterize a rotation matrix as the pseudo rotational vector do) of finite rotation which would itself become additive through the iterative solution process? After get this "total" rotational vector, the final rotation matrix can be recovered from it. The two papers i listed above seemed to have used such a strategy. Or do I misunderstand them?

There are two strategy to update the rotation as shown in the paper attached below. One possesses the additive property indeed! 

Conference Paper A review on geometrically exact models for very flexible rods

In addition to the references included in previous answers to your question, I will add the following, that may be helpful for you:

Ritto-Correa M, Camotim D, On the differentiation of the Rodrigues formula and its significance for the vector-like parametrization of Reissner-Simo beam theory, Int. J. Numer. Meth. Engng. 2002, 55: 1005-1032

The book

Crisfield MA, Non-linear Finite Element Analysis of Solids and Structures. Volume 2: Advanced Topics, Wiley

also includes several chapters devoted to finite element formulations for rods undegoing large displacements/rotations.

Dear Lázaro,

thanks a lot for taking time out to answer this question. References you recommended are classic.