2. Cinematics

2.1. Deformations

2.1.1. Continuous medium

In this case, the displacements of the structure are represented by a vector field \(u\) with three components in general.

The deformation (in the hypothesis of small disturbances) is defined by the deformation tensor \(\varepsilon\) by (options EPSI_ELGA and EPSI_ELNO):

\({\varepsilon }_{\text{ij}}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i})\)

We may want to calculate the « mechanical » deformation, that is to say by subtracting the thermal expansions (options EPME_ELGA and EPME_ELNO):

\({\mathrm{\epsilon }}_{\text{ij}}^{m}\left(u\right)=\frac{1}{2}\left({u}_{i,j}+{u}_{j,i}\right)-{\mathrm{\epsilon }}^{\text{th}}\)

For non-linear calculations, it is sometimes interesting to know the plastic deformation (options EPSP_ELGA and EPSP_ELNO) noted \({\mathrm{\epsilon }}^{\mathit{pl}}\).

In the case of large displacements, the Green-Lagrange deformations are (options EPSG_ELGA and EPSG_ELNO):

\({E}_{\text{ij}}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})\)

From which we may want to subtract thermal deformations (options EPMG_ELGA and EPMG_ELNO):

\({E}_{\text{ij}}^{m}(u)=\frac{1}{2}({u}_{i,j}+{u}_{j,i}+{u}_{k,i}{u}_{k,j})-{\varepsilon }^{\text{th}}\)

For the hypothesis of plane deformations (D_ PLAN), it is important to note that the plane deformation condition is written on the total deformation:

\({\mathrm{\epsilon }}_{33}\left(u\right)=0\)

If we write the total deformation as the sum of a purely mechanical deformation and the deformation resulting from the control variables (such as the thermal expansion deformation):

\({\mathrm{\epsilon }}_{33}\left(u\right)={\mathrm{\epsilon }}_{33}^{m}+{\mathrm{\epsilon }}_{33}^{\mathit{th}}=0\)

So it is natural that in plane deformations, the mechanic deformation out of plane is not zero in the presence of thermal expansion.

\({\mathrm{\epsilon }}_{33}^{m}\left(u\right)\ne 0\)

2.1.2. Case of beams

In traditional beam theories, each \(P\) point on the beam represents a straight section. It is therefore the elements of reduction of the torsor \((T(s),\Omega (s))\) and the displacement of the supposed rigid straight section that characterize the displacement of the point \(P\) to the curvilinear abscissa \(s\). \(T\) is the translation of the center of inertia of the section, \(\Omega (s)\) the rotation vector of the section at this point.

The application of the virtual work theorem (cf. [bib2] _) naturally leads to defining as deformation the torsor \((\varepsilon ,\chi )\) derived from \((T(s),\Omega (s))\) with respect to the curvilinear abscissa \(s\):

\(\begin{array}{}\varepsilon =\frac{\mathrm{dT}}{\text{ds}}+\tau \wedge \Omega \\ \chi =\frac{d\Omega }{\text{ds}}\end{array}\)

So let’s say:

\(\begin{array}{}\varepsilon ={\varepsilon }_{L}\tau +{\gamma }_{T}\\ \chi ={\gamma }_{t}\tau +\text{K}\end{array}\)

\({\varepsilon }_{L}\) is the longitudinal deformation,

\({\gamma }_{T}\) is the vector of the distortion deformations (zero in the Navier-Bernoulli hypothesis),

\({\gamma }_{t}\) is the torsional deformation of the section,

\(\text{K}\) is the bending strain.

Note:

For beam models taking into account warpage, kinematics is more complicated to describe, but they nevertheless lead to concepts similar to those presented above.

2.1.3. Case of shells

Here we will limit ourselves to the cases of plaques. In fact, in the general case of shells:

• spatial derivations use mathematical concepts that are too complicated for the purposes of this document, [R3.07.04],

• shells are very often modelled by elements of assembled shell elements.

In this case, only material normals are assumed to be rigid. The displacement of these normals is therefore represented by the elements of reduction of a \((T,\Omega )\) torsor. \(T\) is the translation of the point located on the middle sheet, \(\Omega\) the rotation vector of the normal at this point.

It is clear that the normal component of \(\Omega\) is zero (in the case of non-micropolar media). The vector \(\text{I}\) is introduced into the tangential plane defined by:

\(\text{I}=\Omega \wedge \text{n}\)

where \(n\) is the normal vector orienting the surface.

That is, the decomposition:

\(T=w\text{n}+{\text{u}}_{T}\)

\({\text{u}}_{T}\) is the tangential displacement,

\(w\) is the arrow.

In the same way as for beams, the application of the virtual work theorem (cf. [bib2] _) leads to defining as deformation the set formed by the tensors \(E\) and \(K\) and the vector \(\gamma\), all these quantities being defined in the tangential plane by:

\(\begin{array}{ccc}{E}_{\alpha \beta }& \text{=}& \frac{1}{2}({u}_{\alpha ,\beta }+{u}_{\beta ,\alpha })\\ {K}_{\alpha \beta }& \text{=}& \frac{1}{2}({l}_{\alpha ,\beta }+{l}_{\beta ,\alpha })\\ {\gamma }_{\alpha }& \text{=}& {l}_{\alpha }+{w}_{,\alpha }\end{array}\)

The deformation is therefore defined by 7 real numbers.

\({E}_{\alpha \beta }\) are membrane strains,

\({K}_{\alpha \beta }\) are the inverses of the curvatures of the deformed middle sheet,

\({\gamma }_{\alpha }\) is the warp deformation vector.

Note:

Again, there is no universal convention and the disparity between conventions is even greater than for the tensors of effort.

Link with the three-dimensional deformation field

Under these conditions, we have:

\(\begin{array}{cc}{\varepsilon }_{\alpha \beta }& \text{=}{E}_{\alpha \beta }+{x}_{3}{K}_{\alpha \beta }\\ {\varepsilon }_{\alpha 3}& \text{=}{\gamma }_{\alpha }\\ {\varepsilon }_{\text{33}}& \text{=}0\end{array}\)

2.2. Related quantities in Code_Aster

2.2.1. DEPL_R and DEPL_C

The quantities DEPL_R and DEPL_C have as components the degrees of freedom of finite element modeling and therefore do not necessarily have only the components of the displacement fields, which are:

DX, DY, DZ

To whom must be added for beams or shells:

DRX, DRY, DRZ

For shells, we need all three components of the rotation vector, because the finite element equation can only be expressed in a global Cartesian coordinate system.

2.2.2. EPSI_R

The quantity EPSI_R represents the deformations of the structure, so it must have, at a minimum, the components:

• deformation fields \(\varepsilon\) of continuous media (as a global reference frame):

  EPXX, EPYY, EPZZ, EPXY,, EPXZ, EPYZ

• beam deformation fields (in the « user » coordinate system for the beam):

  EPXX, GAXY, GAXZ, KY, KZ, GAT

• fields of shell deformations (necessarily as a « user » coordinate system on the surface)

  EXX, EYY, EXY, KXX, KYY,, KXY, GAX, GAY

2.3. Calculation options

2.3.1. Fields EPSI_ELGA, EPME_ELGA, EPSG_ELGA, EPMG_ELGA, and EPSP_ELGA

These are fields containing the deformations at the Gauss points and possibly at the sub-points of the elements.

2.3.2. Fields EPSI_ELNO, EPME_ELNO, EPSG_ELNO, EPMG_ELNOet EPSP_ELNO

These are fields containing the deformations regardless of the modeling for exploitation purposes (printing or post-processing visualization) at the nodes and possibly at the sub-points of the elements.

2.3.3. Fields DEGE_ELGA and DEGE_ELNO

These are fields containing the generalized deformations on the beam or shell elements for exploitation purposes (printing or post-processing visualization) at Gauss points or at the nodes of the structure.