2. Elasticity in large transformations

2.1. Objective

From now on, we propose to take into account large movements and large rotations, a feature accessible by the keyword DEFORMATION =” GROT_GDEP “in the STAT_NON_LINE command. Let us specify right now that we are restricted to isoparametric finite elements (D_ PLAN, C_, C_ PLAN, AXIS and 3D) for which the discretization of the continuous problem does not pose any particular difficulties, cf. [R3.01.00].

For this purpose, it is assumed that the second Piola-Kirchhof stress tensor, \(S\), derives from the Hencky-vonMises potential expressed using the Green-Lagrange deformation \(E\):

\(\mathrm{S}=\frac{\partial \mathrm{\Psi }}{\partial E}\left(\mathrm{E}\right)\)

Let’s also recall the definitions of \(E\) and \(S\). Additional information can also be found in [bib1].

\(\begin{array}{}F=\text{Id}+\text{Grad}(u)\text{}E=\frac{1}{2}({}^{T}\text{}\text{FF}-\text{Id})\\ S=\text{Det}(F){F}^{-1}\sigma {}^{T}\text{}{F}^{-1}\end{array}\)

Such a behavioral relationship, called hyperelastic, makes it possible to take into account large deformations and large rotations from a precise point of view. However, we limit ourselves to small deformations for two reasons. First of all, the behavioral relationship adopted does not have the right properties (polyconvexity) to ensure the existence of solutions and does not control significant compressions either. Second, plastic behavior differs significantly from hyperelastic behavior as soon as the deformations become appreciable. It is for these reasons that we have chosen to maintain the hypothesis of small deformations, thus escaping the controversy of large deformations.

2.2. Virtual work of external efforts: dead load hypothesis

To deal with the problem of calculating hyperelastic structures, we try to write the balance in variational form on the initial configuration. In particular, it is necessary to express the virtual work of external forces on this same initial configuration, which requires the additional hypothesis of dead loads: it is assumed that the load does not depend on the geometric transformation. Typically, an imposed force is a dead load while the pressure is a subsequent loading since it depends on the orientation of the application face, and therefore on the transformation. Under this hypothesis, the virtual work of external efforts is written as a linear form:

\({\mathit{dW}}_{\text{ext}}\text{.}\mathrm{\delta }\mathrm{v}=\underset{{\mathrm{\Omega }}_{o}}{\int }{\mathrm{\rho }}_{o}{F}_{i}\mathrm{\delta }{v}_{i}d{\mathrm{\Omega }}_{o}+\underset{{\partial }_{F}{\mathrm{\Omega }}_{o}}{\int }{T}_{i}^{d}\mathrm{\delta }{v}_{i}{\text{dS}}_{o}\)

\(F\): volume loading

\({T}^{d}\): surface loading acting on the edge \({\partial }_{F}{\Omega }_{o}\)

2.3. Virtual work of inner efforts

We will not give a demonstration of the expressions presented here. For this, we can refer to [bib1] and [R7.02.03]. Again, we choose the initial configuration as the reference configuration, to express the work of internal efforts:

\({\text{dW}}_{\text{int}}\text{.}\mathrm{\delta }\mathrm{v}\text{=}\underset{{\mathrm{\Omega }}_{o}}{\int }{F}_{\text{ik}}{S}_{\text{kl}}\mathrm{\delta }{v}_{i,l}d{\mathrm{\Omega }}_{o}\)

with: \(\delta {v}_{i,l}=\frac{\partial {\mathrm{dv}}_{i}}{\partial {X}_{l}}\)

With a view to resolving by a Newton method, it is important to also express the second variation of the virtual work of internal forces, namely:

\({d}^{2}{W}_{\text{int}}\text{.}\mathrm{\delta }\mathrm{u}\text{.}\mathrm{\delta }\mathrm{v}=\phantom{\rule{2em}{0ex}}\underset{{\mathrm{\Omega }}_{o}}{\int }\mathrm{\delta }{u}_{i,k}{S}_{\text{kl}}\mathrm{\delta }{v}_{i,l}d{\mathrm{\Omega }}_{o}\)	Geometric stiffness
\(\mathrm{...}\phantom{\rule{2em}{0ex}}+\underset{{\mathrm{\Omega }}_{o}}{\int }\mathrm{\delta }{u}_{i,q}{F}_{\text{ip}}\frac{{\partial }^{2}\mathrm{\Psi }}{\partial {E}_{\text{pq}}\partial {E}_{\text{kl}}}{F}_{\text{jk}}\mathrm{\delta }{v}_{j,l}d{\mathrm{\Omega }}_{o}\)	Elastic stiffness

2.4. Variational formulation

We now have at our disposal all the ingredients to write the variational formulation of the problem:

\({\mathit{dW}}_{\text{int}}\text{.}\mathrm{\delta }\mathrm{v}={\text{dW}}_{\text{ext}}\text{.}\mathrm{\delta }\mathrm{v}\), \(\forall \delta v\) kinematically admissible